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MATHEMATICS 


The person charging this material is re- 
sponsible for its return to the library from 
which it was withdrawn on or before the 
Latest Date stamped below. 


Theft, mutilation, and underlining of books 
are reasons for disciplinary action and may 
result in dismissal from the University. 


UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN 
MAY <3 19/0 

A h\ iF noePn 

MAY F ee 


L161— O-1096 











SCHOOL ALGELRA 


FIRST COURSE 


> Leer 


BY 
Pewee RZ Pa. 


UNIVERSITY OF ILLINOIS 


A. R. CRATHORNHE, Pa.'D! 
UNIVERSITY OF ILLINOIS 
AND 


E. H. TAYLOR, Pu.D. 


EASTERN ILLINOIS STATE NORMAL SCHOOL 





NEW YORK 
HENRY HOLT AND COMPANY 
1915 








et ANAL, 
AVMAMMONATCHOVANU 


NAAT 





Da. | USOur. 
Rt 


MATHEMATICS LIBRARY 


PREFACE 


| Tuts book is the first volume of a two-book series. It con- 
| ains ample material for a full year’s work in the first year of the 
aa school, and covers the parts of algebra most likely to be 

of use to the student who goes no further in the subject. 
Tt will prepafe for Plane Geometry and Physics, which come 
in the later years of the high school. The second volume, 
the Advanced Course, supplies the additional material de- 
manded for entrance into the scientific and technical courses 
in our colleges and universities. 

The text represents a special effort at presentation of prin- 
ciples and definitions in clear, simple style, with wordy unes- 
sentials eliminated. The study of algebra is taken up as an ex- 
tomien of arithmetic. The laws of algebra are first suggested 
by induction from familiar rules of arithmetic, and throughout 
the book the close connection with arithmetic is kept in view. 
The pupil is led to see that new symbols are introduced into 
algebra, not arbitrarily, but because of their real advantages 
in representing numbers. 

Difficulties are taken one at a time at sufficient intervals to 
allow the mastery of each one before proceeding to the next. 
Thus the fundamental idea of representing numbers by letters 
is developed in the first two chapters; that of signed numbers in 
Chapter III; while the equation is not formally introduced 
until Eiiater IV, though it is used informally without special 
efinition from the beginning. In accordance with the same 
‘beneral scheme, each of the several chapters on the equation 
takes up a single new difficulty at a time when the pupil is 
ready for it. 








oye | 


A fourth fundamental topic, the notion of a function, is gradu- 
ally approached through evaluation of expressions and rou! 
the graph, so that the pupil is prepared for the more formal treat, 
ment, with the use of the functional notation, in the advancec 
course. Graphical work is treated as simply as possible, not 
as an added difficulty, but when occasion for it arises in th 
algebraic work. While graphical solution is important, it is 
not the fundamental notion underlying graphical representa- 
tion. The idea of functionality, the change in a function as the 
independent variable changes, is the idea about which the 
work in graphs should center. 

The book is well supplied with carefully graded exercises 
and problems. In general, preference has been given to mis- 
cellaneous groups rather than to groups illustrating but one 
process. With regard to the question of correlation with other 
high school subjects the book follows a middle course. Many 
problems involving applications to other subjects have been 
introduced, but care has been taken to keep within the limits 
of the pupil’s experience, as well as to give data that may be 
depended upon to be correct. 

During the last decade many teachers’ associations have dis- 
cussed the arrangement and content of the high school course 
in algebra. Many outlines of courses have been proposed and 
some detailed syllabi have been published. In the preparation 
of this book we have availed ourselves of these discussions and 
printed syllabi, and have endeavored to incorporate into the 
work the views which prevail among progressive teachers. We 
take pleasure in expressing to many teachers our appreciation 
for helpful suggestions and criticisms. We are especially in- 
debted to Prof. E. J. Townsend and Dr. E. B. Lytle of the Uni- 
versity of Illinois and to Miss Jessie D. Brakensiek of Quincy 
High School, Quincy, IIl., for careful and critical reading 
of the manuscript and for suggestions as to exercises and 
problems; to Mr. J. L. Dunn of the Lewis and Clark Higt 
School, Spokane, Wash., Mr. C. H. Fullerton and Mr. W. B.: 


lv PREFACE 


Tt a mg a 


PREFACE Vv 


Skimming of the East High School, Columbus, O., Mr. E. A. 
ook of the Commercial High School, Brooklyn, N.Y., Mr. 
. C. Irwin of the Joliet Township High School, Joliet, IIl., 
nd Mr. R. L. Modesitt of the Eastern Illinois State Normal 
chool, Charleston, Ill., for reading the proof and seeing the 
ook through the press. 






H. L. RIETZ 
A. R. CRATHORNE 
Hi LAN LO 


\ 
{ 
| 





CONTENTS 


NOD ONNN ORR He 





CHAPTER I 
INTRODUCTION 
PAGE 
. Numbers and Language of Arithmetic. .........2.. 
IIEIPERIGOOTA 0 re te kk kk 
Smuprmmriana@r UNeETALION. . ... 6.1. 6 ee ee 
m. Use of Letters in Solving Problems. .......2.2.2.2.~. 
. Use of Letters to Abbreviate Statements ...... Sete 2 
EOS ee a 
Serermerierenra of @ Circle... ww kk kk 
Seenren Ora wircle =... .. 1... . AR ce BM 
PUTS oe kw. Lyne (tk CRE aes ee ere 
oo Se ee eae re SE 3 
CMUMEPIEIEIIIREOWETH 9.0. ec. c e te ee 
Mertmmrerasotoonpymbols ... . . . 1 1 wk ew we 1 
CHAPTER II 
ALGEBRAIC EXPRESSIONS 
MnMMtRIPREROIFOMRIONS, .. . . wk tt ke tk ke le 14 
RIVIERA OMA ok eh oe ee ke ee 14 
SS ee 15 
. Evaluation of Expressions... . . RG My Se coats aed 16 
pereapreseions « ontaining one Letter . . 2... 0.2... .8.. 17 
8B. Graphical Representation. ...... Ee A ek + at aes 19 
CHAPTER III 
POSITIVE AND NEGATIVE NUMBERS 
®. The Use of a Scale to Represent the Numbers of Arithmetic. . 23 
p. Addition and Subtraction on the Scale... .... . Si tte 
pabosmive and Negative Numbers .....:... . Mare ee | 


vill CONTENTS 


22. Illustrations. . ..0 0 6% 3s 5 
23. Numerical or Absolute Value. . .. . Pee 
24. Greater and Less. ..: 2. 0. 4 2 0) 2 a 
25. Addition of Signed Numbers. . . . <2) pige enn 
26. Subtraction of Signed Numbers . . . 0) 3 
27. Subtraction on a Seale... . . . . soe eee 
28. Rule for Subtraction . - .. . . 2 . Se 
29. Addition and Subtraction of Several Numbers. ....... 
30. Multiplication in Arithmetic. - 2 . 293 S52 ee 
31. Multiplication of Signed Numbers . ~ 790) eee 
32. Division of Signed Numbers - . © >.> 25.) ee 
33. Fractions 2... 2... 1s 6 ee se 


CHAPTER IV 
EQUALITIES 


34, Members of an Equality. . . . . . 2 . 290) ye 
35. Identities. 2... 2... 2. eh. Ds 
3G, puquatlons YU - meicdseeeeeh) eee eS . do ke, 
37. Solution of Equations. . . .. .. . . . 29a 
38. Principles Used in Solving Equations : .. 3.) Jee 
39. Verification of Solutions by Substitution. . . 7) 2) ieee 
40. Transposition . . 2... . . ...5 + « 9) 0 
41. Translation of English Expressions into Algebraic Expressions . 


CHAPTER V 
ADDITION 


42. Terms of an Expression. . ... . . « = 2) 0) 
43. Monomials and Polynomials... . . . ) 3 
44, Similar Terms . .-. . . .. . 3-4 « 1) < 
45. Addition of Monomials . ... . . . 9 3933 
46. Simplifying Polynomials. . . . >... + sue 
47. Arrangement of Terms ina Polynomial. .......... 
48. Addition of Polynomials. ... ... . . J 2 


CHAPTER VI 


SUBTRACTION 


49. Subtraction of Monomials. . .. .. . . ss gene 
60... Subtraction of Polynomials... . . issue ke aaa 











CONTENTS 


CHAPTER VII 
PARENTHESES 


CHAPTER VIII 
MULTIPLICATION 


#. Multiplication of a Product by any Number... ...... 
fetrouuc, Oo. &tolynomial by a Monomial .......... 
merit !wo Polynomials... 2 .. . . 2. 2 ee ee 


CHAPTER IX 
EQUATIONS AND PROBLEMS 


pebeuations lrvolving Parentheses... . i... ...:. 66656 e-e 
mrouemone involwing Fractions... . . ... 2 6. «se ew 


CHAPTER X 
DIVISION 


SRE OOMUAI 3 ec ee ke 
Division of a Polynomial by a Monomial .......... 
erry me OrvPOUNaL 5 6 kk kk ee 
SR co 


CHAPTER XI 
LINEAR EQUATIONS 


NMEETIREUQUIS Eo ge a sk i emf eee ee el 


CHAPTER XII 
IMPORTANT TYPE FORMS 


EN ee) es ee wei. @ gale ale 


MMII ICAL cg my ge se ee es 


1X 


72. 
73. 
74. 
75. 
76. 
77. 
78. 
79. 
80. 
81. 
82. 


83. 
84. 


85. 
86. 
87. 


88. 
89." 
90. 


CONTENTS 


. Product of the Sum and Difference of Two Numbers. ... . 
. Product of Two Binomials Having a Common Term .... . 


Cube of a Binomial... 3... 3 Le 


- Square of a Trinomial, © . <>... > 7) Ree 


CHAPTER XIII 
FACTORING 


Prime Factors in Arithmetic. . 5...) 
Prime Factors in Algebra . . .°.. . 40 2 ee 
Factors of Monomials. . Sa 

Monomial Factors in Polynomials 

Factors Found by Grouping. Terms. . . © .. 2m 
Difference of Two Squares 

Trinomial Squares .. . « at! he ah ee 
Trinomials of the Form 2? a ay e ate x x ae . igo ae os 
General Quadratic Trinomial. . . . . Mer 
Sum and Difference of Two Cubes... . . 2) Se 
Summary of Factoring ...-. .. ..% . - ne 


CHAPTER XIV 
EQUATIONS SOLVED BY FACTORING 


Quadratic Equations . 
Factoring Applied to the Bolitian Ef Guanrane Equationes 


CHAPTER XV 


HIGHEST COMMON FACTOR AND LOWEST COMMON 
MULTIPLE 


Greatest Common Divisor in Arithmetic. . . . . jae 
ae Common Factor oe Sk WS Ak 


CHAPTER XVI 
FRACTIONS 


Fractions in Arithmetic... ... ... + . ee 
Fractions in Algebra ..... . ere 
Division by Zero... 6.0. «6 6 se 8 














} CONTENTS xi 
{ PAGE 
Po Oe STS Sl Pe 138 
( Reduction to Lowest ane Pe aati 0 Fe Oe | ee ere 140 
Je Gancellation...... . Oo Loe! ee pes TAl 
: « Reduction to Common Dencnistor SOL Se ea 
(. Addition and Subtraction of Fractions... ......... 148 
i Beoneamomorrractions 2... wk ee eh 1G 
PMIMOSEOUETAGUONS 6 ee ee 1D 
, I SE Gr a ee ks Sa ee a 151 
ih 
i| CHAPTER XVII 
FRACTIONAL AND LITERAL EQUATIONS 
ieeeaearing quations Of Fractions. . ........ . '.. + 159 
|0. Unknowns in the Denominator... ........... 160 
SESS ee a a 164 
Sema I MOS Ss pe tk A he ew we ws 165 
6 CHAPTER XVIII 
[ RATIO, PROPORTION, AND VARIATION 
eee Tatag)). ao Ys een est eee ee. F218 
WEIRUECMOTO OAM ww. kk kk _ OEE Sree oy rack dt 
| Mean Proportional”. ..... ae ot oF ae SY Ga eee 1b | 
Perera our troportional. . . ... .. . ...... <1 
Debra Ve UCrnation fos 4... ee ew ee ew NT 
re GrvoMnvernsion ©. ee ee on NTS 
DtrruOnre A OMpodilioOn. - . . . 5. we ee... 8 
. Proportion by Division. . . . eres, * haaiaermmrmemte eats WE | 
. Proportion by Composition and Divison: Oey. | abery > einen be 
fo DI SONRtANIS ws ee et ee eS TE 
fa SE, Oe a a TG Wy 5 
i CHAPTER XIX 
| 
| GRAPHICAL REPRESENTATION OF THE RELATION 
i BETWEEN TWO VARIABLES 
4, I 2 a es we ete 180 
| i Axes, Codrdinates. . .. . Pea Pe A Pee) Aye nek st LO 
SMIIPRIRMEL Ce hy es ks ces eee Loe, 


ES RR RD, Cee an gee aire Wiki 183 


Ke MOTTO ANCE cin a te ere we ew 182 
( 
Ua INCION ee eG el aw ke 188 


( 


x CONTENTS ; 
PA 
68. Product of the Sum and Difference of Two Numbers... . . , 
69. Product of Two Binomials Having a Common Tle@Rior icra 
70. Cube of a Binomial... 9. .-. . 2) 
71... Square of a Trinomial. . . ..°. | 1p 
CHAPTER XIII 
FACTORING 
72. Prime Factors in Arithmetic. . . >.) Geen 
73. Prime Factors in Algebra . . . \. ) .) a) 
74. Factors of Monomials. . .. . 1 8 6 a or 
75. Monomial Factors in EAlvromiale on ee al 


76. 
“ie 
78. 
79. 
80. 
81. 
82. 


83. 
84. 


85. 
86. 
87. 


88. 
89." 
90. 


Factors Found by Grouping. Terms : <> =) 3) eee 
Difference of Two Squares ......... 

Trinomial Squares .). .°s 5. . .0 5 ee 
Trinomials of the Form a? + Bae + b) x + ab 
General Quadratic Trinomial. . ... » 2 ee 
Sum and Difference of Two Cubes . . . . 3) ene 
Summary of Factoring . .-. . ©. +. Jens 


CHAPTER XIV 
EQUATIONS SOLVED BY FACTORING 


Quadratic Equations... . : 3... 02 
Factoring Applied to the Solin ot Quadratic pes 


CHAPTER XV 


HIGHEST COMMON FACTOR AND LOWEST COMMON 
MULTIPLE 


Greatest Common Divisor in Arithmetic. . . ....4.5.5., 
Highest Common Factor Pee Lge 


CHAPTER XVI 
FRACTIONS 


Fractions in Arithmetic. ... . . . . . ee 
Fractions in Algebra . . . . . . = ws) wen atne nnn 
Division. by Zero... 2. 605 se se 





CONTENTS 


. Signsin Fractions ....... Sipl<_ + .. 7 sera ee ea 


tetcrias Gwent LeIm0s . . . 4... ... « 6 8 bbl eoa a 
@eeaneenation. .. .- .. . fot 3. Se 








Reduction to Common eh arniiator Stee 
mniiog an sultraction Of Fractions. .. . . . ... 9s © ws 


. Multiplication of Fractions 
. Division of Fractions . 
. Complex Fractions . . 


CHAPTER XVII 
FRACTIONAL AND LITERAL EQUATIONS 


. Clearing Equations of Fractions . 

. Unknowns in the Denominator . . 

BRUCPPNAGI ek ee ke ee si 
SI IICEIE ST MUOM GS rcs ss ts, se Ne bape vee 


CHAPTER XVIII 
RATIO, PROPORTION, AND VARIATION 


RS ge >) See nee : 
Se a RS 
EM TMOPUIG GRIER Se ee ee 
eerrraricerouren troportional .. . . 3... 2s ek te ee 
. Proportion by Alternation ..,.. selene re” kOe ome 


Proportion by Inversion . . 
Proportion by Composition. . 


. Proportion by Division. : ee: 
. Proportion by Composition and ivisions het ee eae 
. Variables and Constants . . 


DITA EMVN ST ek ke 


CHAPTER XIX 


GRAPHICAL REPRESENTATION OF THE RELATION 
BETWEEN TWO VARIABLES 


SG eal TS RR 
Axes, Codrdinates. . .. . ria 85 5 8 ue 
POOL OImts’.. 5... - - GM arene case 


eereeuooriimate Paper... 5 6. 8 ee we ke 
OEMS ee Mee OTA ies, eek 


Xl CONTENTS 


120. Graph of a Function. . . .°. 9... 2) ee 
121. Graph of an Equation : . . . . . 12 
122. Locus of a Linear Equation. . . . “4 5) een 
123. Graphic Solution of Equations: . . . .) 73a neuen m. 
124. Graphical Representation of Scientific Data ee ee 


CHAPTER XX 
SYSTEMS OF LINEAR EQUATIONS 


125. Solution of Equations in Two or More Unknowns. .... . 
126. Simultaneous Equations . ... . 2 -, . 9 
127. Independent Equations. . . . . =. = = Secsnneeneenuene 
128. Dependent or Equivalent Equations. ........... 
129. Inconsistent Equations. . .-. .. 72 .) sees 
130. Elimination. . . . ) ... 2. . 4 . 0) Bes 
131. Elimination by Addition ae Siibtracian oe SE a ales 
132. Elimination by.Substitution®’ | . . . 7 | ee 
133. Standard Form az + by +c=0 . . . |. =e 
134. Literal Equations Containing Two Unknowns. ....... 
135. Linear Systems in Three or More Unknowns ........ 


CHAPTER XXI 


SQUARE ROOT AND APPLICATIONS 


246. Definition of a Square Root... . . . 2). . eee 
Sage Hadical Sign wa. . 6 i oe eg eh yee 
138. Square Root of Monomials . . . . . . . sss 
139. Equations Solved by Finding chee are Roots’... [2 aan 
140... Square Roots of Trmomials. . . : ./. 2) 
141. Process of Finding the Square Root . . . . . Jes 
142. Square Roots of Numbers Expressed in Arabic Figures . ; 
143. Explanation of Process of Finding Square Root in Arithmetic : 
144. Number with More than Two Periods. .........°. 
145. Squaré Roots of Decimals ... .. . . 3) 2 0) eee 
146. Approximate Square Root... . : Se) 2) se 


CHAPTER XXII 


RADICALS 


147, Radicals ... . . . 1.05. 6 5 202. Wr 
148. Rational and Irrational Numbers . . <2) 2) eee 2 








70. 
TA: 
72. 
73. 
74. 


CONTENTS 
I ee ek ee eae tg 
Square Root of a Fraction ~ LD MU ice: ur eek 
Bremineniomor Radicals. .-. 6 6 6 ee ee ee 


Meaning of Simplification of a Radical... ... 1... . 
Addition and Subtraction of Radicals *. ......... 
Multiplication of Quadratic Surds .“ . 2 2. 1 ee ee es 
Mivision of Quadratic Surds. . .“.. 3’... se we 
Rationalization of Denominators “: ........... 
MUERTE CtOY fo ee i of ce ee et 
Solution of Equations Involving Reticnls. fo. Ts oh 


CHAPTER XXIII 


QUADRATIC EQUATIONS 


Quadratic Equations Solved by Factoring ........-. 
IEE QUaATC. 2... we ee ee ee es 
Equations Solved by Completing Square. . .......- 


. Solution by Hindu Method of Completing Square... . . . 


Type Form of a Quadratic Equation... .... . 


. Quadratic Solved by Formula. ........- a ee ais 
. The Special Quadratic az? +c=0 .... 2.2. + ee ees 
. Graphs of Quadratic Functions... . . 
MIIPIGREVONUIMDEIS. . . 6 ee es 
. Graphical Meaning of Imaginary Roots ...... . 


Historical Note on Quadratics. . ... . 22 6 1 se ee ee 


CHAPTER XXIV 
SYSTEMS OF EQUATIONS INVOLVING QUADRATICS 


ye yh ESS SS Sa a PRA 
Solution of Simultaneous Quadratics. . . . - 1. - ee. 
One Equation Linear and One Quadratic. . . ...... . 
Equations Containing x? and y2only. .......-.---: 
ES de i 


Xill 
PAGE 
226 


227 


227 
228 
229 
231 
232 
233 
234 
235 


238 
239 
239 
240 
242 
242 
245 
247 
249 
250 
255 





SCHOOL ALGEBRA 
‘FIRST COURSE 


CHAPTER I 
INTRODUCTION 


_ 1. Numbers and language of arithmetic. In counting, the 
child learns numbers which are called integers. 

The written language of arithmetic uses the numerals 0, 1, 
2, 3, 4, 5, 6, 7, 8, 9, to represent numbers and the signs +, —, 
&, + to denote operations. 

In problems where two or more integers are added or are 
multiplied together, or when the smaller of two integers is sub- 
tracted from the greater, the answer is always an integer. In 

he use of the sign + however, another kind of numbers, called 
ractions, is obtained. These two kinds of numbers, integers 
and fractions, have been studied in arithmetic. 

| @, Language of algebra. Algebra is a continuation of arith- 
metic. ‘In it we use not only the numbers and symbols of 
arithmetic, but we also introduce new kinds of numbers and 
new symbols. The written language of algebra makes much 
use of letters to represent numbers. This use of letters is not 
entirely new to the student, for it is customary in arithmetic to 
represent certain numbers by letters. 


Thus, the radius of a circle is often represented by r, the diameter by 
d, or by 2xr. The altitude and the base of a rectangle may be repre- 
sented by a and b, the area by a Xb, and the distance around by 
at+ta+b+b, Bee 2 xb. 


3. Symbols of operation. The signs +, —, X, + are used 
‘in algebra as in arithmetic, but the sign of aot is 


I 
| ] 
2 INTRODUCTION [Cuap. I 





usually omitted. For example, ax 6 is usually written ab 
| and 2x ris written 2r. If a sign of multiplication is used, it i 
customary to use a dot, written a little above the position for 
a period to distinguish it from the decimal point, instead of 
\the sign x. For example, 2 x 3 is written 2:3. The sign of 
division is not much used in algebra. Thus, a+ 6 is usually 
written : 

The use of letters to represent numbers enables us to write 
many statements in very brief form. Thus, if A is the area, b 
the base, and a the altitude of a rectangle, the brief statement 


A =ab 


gives the rule that the area of a rectangle is equal to the prod 
uct of the base and altitude. 

It is important to be able to translate English sentences into 
such algebraic statements. : 





EXERCISES 


1. If / stands for the length of a running track in yards 
what stands for the length of a track 50 yards longer? aa 
1+ 50 yards. 

2. If l stands for the length of a track, what is the leratif 
of a track 100 yards longer? What is the length of a tracl 
twice as long? 

3. What is the cost of 5 railway. tickets at a2 dollars a 
ticket? 

4. What is the wale of 3a when a is 1? When a is 2? 
When a is 13? 

5. How many pecks are there in 6 bushels? Inz bushels? 

6. How many inches are there in y feet? : 

7. If n represents a number, what represents a number 
three times as large? 

8. Write the sum of a, b, c, and d, using the sign of addition. 

9. Write the product of a and 2 as it is expressed in algebra 


, 





{ | 


Ane, 3] LANGUAGE OF ALGEBRA 3 


10. Why is it not good form to write 25 for 2x 5? 

11. Write the following without using a sign of multiplica- 
mion: S8x2z,bxX6,7xaxb,xxyxz,5xrxt,Ixmxnxv. 

12. Indicate the subtraction of + from a By using the sign 
of subtraction. 

13. Write the product of a, b, c, and din tres different ways. 

14. If a is an integer, what is the next integer? What is 
the preceding integer? 

15. A barrel contains 313 gallons. If cis the number of 
cubic inches in a gallon, what is the number of cubic inches 
in a barrel? 

16. If x represents a certain number, what represents a 
number 10 greater than twice x? 

17. If p is the cost of one article, what is the cost per 
dozen? 
| 18. If d is the cost per dozen, what is the cost of one? 

19. What is the simple interest on $100 for three years at 
6 % per annum? On z dollars for y years at m per cent? 
| 20. If x and y are the lengths of two lines, what is their 
combined length? 
| 21. The age of a father is 30 years more than twice his 
son’s age. If x is the son’s age, what is the father’s age? 

22. A rectangle is twice as long as it is wide. Let x be its 
width. What is its length? Its perimeter? Its area? 
93. A rectangle is 8 feet wide. If x is the length, what is 

the area; the perimeter? 

24. Write in algebraic language the statement that the 
sum of two numbers, a and ), is divided by the number c. 
25. Five times 6 plus three times 6 plus two times 6 equals 
how many 6’s? 

26. 5a+ 3x+ 2x = how many 2’s? 

27. 64+ 2a+a=? 
i> 98. 8¢—474+ 27+ 5c =? 
| 99. y+ by—3yt+hy=? 

30. Oz)}+ $2 — 2+ g2)+ 62 — 22 = ? 


{ 


4 INTRODUCTION [ Cuap. 


Fill out the blanks in the following : 
31. 1b+ 5b-—4b—-—b=() 0b. 
32. 30+ ( ) a bt=— 07, 

33. ( )a+3a— 2a— $a = 4a. 


4. Use of letters in solving problems. The following 
examples show the further use of letters to represent numbers 
and to simplify the solution of certain problems. 

Example 1. The sum of two numbers is 128. The larger is 3 times 
the smaller. Find the numbers. 


Let n = the smaller number. 


Then 3n = the larger number, | 
and n+ 3n = 128, the sum. b 
Adding, 4n = 128. 
Dividing by 4, = = 32, the smaller number. be 
Then 3n = 96, the larger number. @ 


Example 2. A lot is sold for $720, which is 20% more than it cost 
What was the cost? 5 


The following is a common form of the arithmetical solution. 


Let 100% of the cost = the cost. 


Then 120% of the cost = HEY. . 
1% of the cost = 745 of $720 = $6, 
and 100% of the cost = 100 x $6 = $600. 


Hence, the lot cost $600. b 


The solution may be shortened by the use of letters. 
Let c = the number of dollars the lot cost. 
Then 1.20c = 720, 
and c = 720 + 1.20 = 600. 
Hence, the lot cost $600. 
Note that c is a number and is not the cost. 


EXERCISES AND PROBLEMS 


1. A house and lot are worth $6000. The house is worth 
four times as much as the lot. What is the value of the lot? 

2. During a January cold wave the price of soft coal in- 
creased 10%. At the end of the cold period the price was | 
$3.85 per ton. What was the price at the beginning? 


i 


i 
Arr, 4] USE OF LETTERS IN PROBLEMS 5 


3. A book sells for $2.40. The dealer makes a profit of 
‘25% of the cost price. What was the cost of the book to 
the dealer? 

4. The sum of the edges of a cube is 48 inches. What is 
the length of one edge? 

5. A merchant sells a hat for $3.50, making 25% profit. 
What did the hat cost the merchant? 

6. For what number does xstand in the equality 3 +2 = 13? 

Hint: Here z is the number which added to 3 gives 13. 

In the following exercises, the letters stand for unknown 
numbers. Find the numbers. 


ih Cae Aas 15. 2 — 20 = 90. 

8724 10= 12. 16. 12n — 5n+ n= 64. 
9. 22=24. | 17. «+30 = 75. 

10. 407 = 28. LB oyetG. 

11. 2—5=0. 19. w—3.= ll. 

12. 52 = 10. 20. 2+5=842. 

13. 7y + 5y = 120. 21. 2+ 22+ 32 = 12. 
14. 3w = 21. 


22. Think of a number; double it; add 15. If the result 
is 52, what was the first number?. 

23. If a certain number is multiplied by 4, the result is 5 
more than 33. What is the number? 

24. A boy bought 3 books. One. book cost twice as much > 
as the other two put togethers If these two books were the 
same price, and if he received 4 cents change out of a dollar, 
what was the price of each book? 

25. Two men, Smith and Jones, do a certain piece 
of work for $300 at $2.00 per day each. Smith works 5 
times as many days as Jones. How many days did each 
work? 

26. In a fire, Smith lost 3 times as much as Jones, and 
Brown lost 4 times as much as Jones. If the combined loss 
was $5120, what did each lose? | 


6 INTRODUCTION | [Cuar. I. 


27. The greater of two numbers is 4 times the less. Their 
sum is 240. What are the numbers? 

Hint: Let » equal the smaller number. 

28. The sum of two numbers is 264. The greater is 11 
times the less. What are the numbers? 

29. The sum of three numbers is 54. The second is twice 
the first, and the third three times the second. Find each 
number. 

30. The perimeter of a rectangle is 216 feet. The rectangle 
is twice as long as it is wide. What are its dimensions? 

31. A man buys 4 books costing x cents each, and 2 balls at 
x cents each. The whole cost one dollar and a half. What is 2? 


do. Use of letters to abbreviate statements. As pointed 
out in Art. 3, the use of letters to represent numbers enables 
us to write many statements in very brief form. A noticeable 
example is the use of letters to abbreviate the rules of arith- 
metic. In Art. 3, the rule:— The area of a rectangle is equal 
to the product of the base and the altitude —is stated in the 
algebraic form . Ae ab, 


in which A represents the area, a the altitude, and b the base 
of the rectangle. | 

Similarly, if we were given the length /, the width w, and 
the height h, of a rectangular solid, the rule for finding the 
volume is expressed in the form 


V = lwh. 


. 


The rule in arithmetic for the product of two fractions is: | 
The product of two fractions is a fraction having for its numera- 
tor the product of the two numerators, and for its denominator © 
a 


the product of the two denominators. If we let F 


and : repre- | 


sent the two fractions this rule becomes 


Ants. 6,7,8] AREA OF A TRIANGLE 7 


6. Area of a triangle. In arithmetic it was learned that 
the area of a triangle is equal to half the product of the base 
and altitude. In symbols 


bh 
A= 


where A is the area, b the base, 
and h the altitude. 

7. Circumference ofa 
circle. In arithmetic it was 
learned that the circumference 
of a circle equals nearly 3.1416 times the diameter. The 
number 3.1416 is approximately the ratio of the circumfer- 
ence to the diameter and is denoted by the Greek letter 7 
(pronounced pi). In symbols 


b 
Fia. 1 


C= 71d, 


where d is the diameter and c the circum- 


poet ference. 
% 8. Area of a circle. The area of a circle 
is one half the product of the circumference 
Fig. 2 and radius. Thus, 
cr 
A r= 9” 


where c is the circumference, 7 the radius, and A the area. 


EXERCISES AND PROBLEMS 


1. If A is the area, x the base, and y the altitude of 

a triangle, write in algebraic symbols the rule for finding the 

| area. | 

2. The base of a triangle is 40 inches and the altitude is 
60 inches. Find the area. ) 

3. A room is 18 feet long, 14 feet wide, and 8 feet high. ~ 

How many cubic feet of air are there in the room? 


8 INTRODUCTION [Cuap. I. 


4. State in algebraic symbols the rule for the length ¢ 
of the circumference of a circle when the radius r is 
given. ; 5 

5. State the rule for the area of a circle when the radius 

r and the circumference c are given/’ 

6. What is the interest on $300 for one ea at the rate 
of 6 % per annum. 

7. Write in algebraic symbols the rule for finding the 
interest on a sum of money for one year. Let J represent the 
interest, P the sum of money in dollars, and r the ae on a 
dollar for one year. 

8: What is the simple interest on $300 for 4 years at 6% 
per annum? 

9. If P is the principal, r the rate, and n the number of 
years, write in letters the rule for finding the simple in- 
terest. } 

10. State in words the rule of arithmetic for finding the 
quotient of two fractions. Bigs this rule into algebraic 
; as the dividend and © 74 the divisor. 

11. If we know the sum of two numbers and one of the 
numbers, how may we find the other? State the rule in alge- 
braic symbols. Let s be the sum, a the known number, and a 
the unknown number. 

12. State in algebraic symbols the rule for finding one of 
two numbers when their product and one of the numbers are 
given. | 

13. Write the rule for finding the altitude a of a rectangle ~ 
when its area A and its base 6 are given. : 

14. The volume of a brick is 64 cubic inches. Its length is 
8 inches, its width 4 inches. What is its thickness? 

15. Write the rule for finding the thickness ¢ of a brick, 
when its volume JV, its length /, and its width w, are given. i 

16. Write the rule for finding the dividend D, when the » 
divisor d, and quotient g, are known. | 


symbols using 


ers, 610/911 | FACTORS 9 


9. Factors. Each of two or more numbers whose product 
‘Is a given number is called.a factor of the given number. 

Thus,;in 2-3 = 6, 2 and 3 are factors of 6; in 2°3° 5 = 80, 
2, 3, 9, 6, 10, and 15 are factors of 30. Similarly, 5, 2, y, 52, 
5y, and «zy are factors of 5xy. 


10. Coefficient. If a number is a product of two factors, 
then either of these two is called the coefficient of the other in 
the product. 

Thus, in 2°3, 2 is the coefficient of 3, and 3 is the coefficient 
of 2. In 3az, 3 is the coefficient of az, 3a of x, x of 3a, and a 
of 32. 

The numerical coefficient 1 is usually omitted. Thus, 

Lh Os 
In such expressions as 3az, the factor consisting of Arabic figures is 
often called the coefficient. 
| EXERCISES 

Give factors of 12; of 14; of 21; of 386; of 120. 
Write six factors of 2ab. 
Write ten factors of 3abc. 
Write twelve factors of 62xyz. 
What is the coefficient of x in 6ay ; of a in fa. 
. Write down factors of the following and give the coeffi- 
cient of each factor. 72; 6a; Sry; 8ab; ary; 9xz; abc; 
12abc. 


Dorm wp 


11. Exponents and powers. An exponent is a number 
written at the right of and slightly above another number 
called the base. When the exponent is an integer, it shows how 
often the base is taken as a factor. 

Thus, 
ae 424° 6 = 5-5-5, 
Pode Oe y= Bes yy. Y. 


A power of a number is the product obtained by using the num- 
ber as a factor one or more times. 


10 INTRODUCTION [Cuap. I. 


ce 


We may read a™ as “‘a exponent m” or the “mth power 
of a.” Since 16 = 47, 16 is ‘4 exponent 2,” or the “second 
power of 4.’’ The number 2° is read “‘2 exponent 5” or the 
“fifth power of 2.” 

When the exponent is 1, it is usually omitted. Thus, a! =a. 
When the exponent is 2, ans power is called the square of the 
base, and when the exponent is 3, the DO is called the cube 
of the base. ee a? is usually read ¢ ‘a square,” and a* is 
usually read ‘‘a cube.” | 

The meaning of an exponent that is not one of the integers, 
1, 2, 3, . . . will be explained later in the course in algebra. 


"EXERCISES 


Write the following products using exponents. Read the 
answers aloud. 


Lid 2 de 2: 130 G+ oe 

PERS Fes eB 3 8. 8-a-a@- 2. 

Of) carey 9. 2 2 - eee 

4. ‘27. 10. 2-2-2 Gee 

Bis eat 11. 81-a-a-272epem 

6. 7-3:-3-3:3. 12. 125-2-2-2-o aoe a. 
Write the following without exponents. 

13. 6°. 16. 72. 19. (4). 22. a’. 

14. 3°. ii 20. (.02)4. 23. "274". 

15. 43. 18/325 21. (2)3. 24. abc. 


~ 


If a= 2, b= 3, and n= 5, find the numerical value of the > 
following. 


25. a’. 28. 6". 310% 34. an. 
2G ome: 297 orb. 32. S7o: 35. 37b?n?. 
27. 2". 30. ‘BF. 33. 2'n". 86. Gab? + a3. 


37. What exponent is understood when none is expressed, . 
as in a, or 2, or ax? 


“Arr. 11]. PROBLEMS 


11 


MISCELLANEOUS PROBLEMS 


1. The second of three consecutive even integers is n. 


What are the other two numbers? 


2. How many square inches are there in a rectangle x 


feet wide and y feet long? 


3. A brick whose height is x inches, is twice as wide as it 
ds high, and twice as long as it is wide. What is its volume? 
| 4. I take twice as long to walk from my house to the 
station uphill as to return downhill. If it takes 9 minutes to 
walk there and back, how long did I take each way? 


In the following the letters stand for unknown numbers. 


i 


Find the numbers and check the results. 


bare o> = 11. 10. 
6. 14=27+ 10. 1 
7. 84+2=5+0. — 12. 
82 27-= 21: 13; 


9. 2n-+- n= / — 3.6. 


14. A certain number is multiplied by 3. 


is 42. What was the original number? 


Se AN A 4a 
2y= 5+ 3. 
Adz = 5. 

7y + 3y = 1. 


Twice the result 


15. Find a number such that if twice the number be added 


to three times the number, the sum is 100. 


16. The sum of two numbers is 60. The greater is 3 times 


the smaller. What are the numbers? 


17. Four times A’s money is $3000 more than B’s. B has 


$8400. How much has A? 


18. A man made a will leaving $10,000 to be divided among 
3 daughters and 4 sons. Each daughter was to receive twice 
as much as each son. What did each son and daughter receive? 


* Some computers prefer to write 0.15 in place of .15. The form 0.15 
places emphasis on the fact that the integral part of the number is zero 
and removes the decimal point from a place in front of the number to a 

‘position where it is not so likely to be carelessly omitted. In writing 
decimals we shall sometimes use the one form, sometimes the other. , 


12 INTRODUCTION [ Crap. I. 


19. A man bought a number of baseballs, some at 75 cents 
each, and twice as many at $1.25 each. He paid $19.50 for the 
lot. How many of each did he get? 

20. A boy has $2.75 in dimes and quarters. He has 3 times 
as many dimes as quarters. How many of each has he? 

21. Which would you rather have, 3x+ 8y dollars, or 
5a + 6y dollars (a) if « = 1000, and y = 800? (6) If x = 500, 
y= 500? (c) If x = 600, y = 700? 

22. Write in symbols: The sum of the squares of a and b 
divided by the cube of c. 

23. Write the powers of 10 from the first to the tenth 
power. 

24. The number 343 is ati power of 7? 64is what power 
of 8? Of 4? Of 2? 

25. Write as a power of 12 the number of cubic inches in a 
cubic foot. 

26. Write ten factors of 8a?2’. 

27. Write in symbols the weight of a rectangular tank of 
water. Let a, 6, c be the dimensions of the tank in feet, and w 
the weight of a cubic foot of water. 

28. Using the answer to Problem 27, find the weight of a 
tank of water, for which a = 2, b = 3, c= 10, and w= 62.5. 

29. A man saves 2 dollars the first year. During the second 
year he saves 100 dollars more than in the first year. In the 
third year he lost half of his savings for the first two years. 
What were his net savings for the three years? 

30. If V is the volume of a cone, b the area of its base, and 
h the height, it is known that 


Give this result in words. 


12. Historical note on symbols. The history of the early use of 
mathematical symbols is very interesting, and shows how mathematical 
progress was retarded on account of defects in symbolism. To appre-» 
ciate in a slight way some of these defects, we may well think of doing a 


‘An. 12] HISTORICAL NOTE ON SYMBOLS 13 


fairly long calculation with Roman numerals I, V, X, L, C, D, M. The 
so-called Arabic notation that uses the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 is of 
‘Hindu origin, the Arabs having obtained it from the Hindus. The great 
‘achievement of inventing a satisfactory method of writing numbers was 
not accomplished in a short time. It required centuries to perfect this 
elegant system. While the dates of the advances are in doubt, it is prov- 
able that the system was complete as early as the sixth century of the 
‘Christian era. 

The basic idea in the system is that of assigning a place value to a 

‘digit. A symbol for zero is necessary to the application of this notion; 
but the importance of a symbol for zero was not recognized until long 
‘after symbols were being used for other numbers. After the principle 
of a place value was established, several sets of characters were used. In 
one of the systems, the circle O was used to denote one and the dot - was 
used to denote zero. It is held by some that the symbols used for digits 
fwere perhaps the first letters of early numerals, and that letters were 
thus used to denote numbers in the earliest attempt at a notation. 
The symbols +, -, X, -, +, = came into common use in the first 
half of the seventeenth century, and their origins are accounted for in 
various ways. For example, it is thought by some that the sign + came 
from the rapid writing of the letter p in the word plus, and by others that 
‘it originated in warehouses in the marking of boxes that showed over- 
‘weight. Thus, if a box, supposed to weigh 50 pounds, weighed 55 pounds, 
it was marked 50+ 5. Whatever may be correct about the origin of 
these symbols, it is doubtless true that previous to their use, the words 
‘for which they stand were generally written out, and it is clear that the 
symbols add greatly to the brevity and elegance of our statements involv- 
‘Ing numbers. 





CHAPTER, if 
- ALGEBRAIC EXPRESSIONS 


13. Algebraic expressions. In algebra, an expression is 
a symbol or combination of symbols that represents a number. 
Thus, gt and x — 2y+ 8z are expressions. If g = 32.2 and 
| t= 10, the first has the value 322. If x=7, y= 2, and z=@ 
the second has a value 15. For different values of the letters, 
the expressions usually represent different numbers. 


EXERCISES 


If a=1, b=2;c=5, x = 4, and y = 2, find the value a 
each of the following algebraic expressions: 


1. a+ 2b+4 3¢. 5. a+ 2bx + dy. 
3b : 
| meh a 
2. dab + be. 6. abay + Da ’ 
3. Sabc + 6a. 7. ab— xy. 
4, 2ab + 3ac + 4be. 8. Zab + Say + be — 3az. 


9. Find the value of 3a+ 2ab— bc, when a=1, b = 2, 
endic = 2: 1 
10. Find the value of 3a+ 2ab— bc, when a=, b= a 
anece =o. a 
iG © ae the value of 3a+ 2ab— bc, when a=7, b= i, 
and c= 4 


14. Order of operations. If, to find the value of the expres- 
sion, 4+ 5-6, we perform the operations from left to right as | 
we come to each symbol, we obtain a result 54. If we perform 
the multiplication first, we obtain the result 34. This simple 
example shows that results depend upon the order in whi 
operations are performed. 

14 





i 
i 

4 

J 


‘Ants. 14,15] | ORDER OF OPERATIONS 15 


__ When nothing is said to the contrary, it is understood that 
in a series of operations involving additions, subtractions, 
multiplications and divisions, the multiplications and divisions 
are to be done in the order from left to right before any 
additions or subtractions. Then additions and subtractions 
are to be performed in any order. 
Thus, | 
2056-4450 = 34, 

and 
25 —-2-6-—442417 = 25 —- 12 —-2417 = 28. 


EXERCISES 


Carry out the indicated operations. 


1. 84-2-34+4+2. 6. 16—4-2-3. 

2. 56+ 2-—2-4+6-+ 2. Gre 2. 2 ee 

3. 18+ 9 + 2. Te AO 42 
4. 18+4-3+ 2. 8. 16-4+2-—4.-2. 
9. xyz — 42+ 2y — 32 for z= 8, y= 2,2=1. 


15. Use of parentheses. ‘To group the parts of an alge- 
braic expression together, parentheses ( ) are ordinarily used. 


Thus, | 
3x + 5y + (w+ y) 
means 

ox + Sy + E+ Y. 


When one pair of parentheses occurs within another, it is con- 
venient to use different forms as follows: [ | called brackets ; 
{}called braces; and _—_— called a vinculum. These signs 
are often called signs of aggregation, but all have the same 
meaning and may be called parentheses. 

In the simplification of expressions involving parentheses, it 
is best to simplify first the expression within the parentheses. 
To illustrate, the removal of the parentheses from 8 + (4 — 2) 
gives 8+ 2 or 10. The expression 2(5+ 3) means 2 times 8 ; 
a(b +c) means the product of the number a and the number 


16 ALGEBRAIC EXPRESSIONS [Cuap. Il. 


obtained by adding together 6 and c. The expression 
(a+ b)(e+d) means the product of the sum of a and b and 
the sum of ¢ and d. 

If in (a+b)(c+d), a=2, b=3,,c= 5, d=6, we have 
(24°3)(5 46) 295 lle bon 


EXERCISES 
Simplify the following : 

1. 10+ (44 2). 

2. 7+ (5+ 2). 

3. c+ (d+ 2c). 

SoLUTION: c+ (d+ 2c) =c+d+2c=d+ 3c. 

4. (a+b)+(c+b). 

5. x +3(5— 2). 

6. «+ 5(7 — 4). 

7. (18 — 2) + (6— 2). 

8. (8 — 4)(16 — 6). 

9. (7+ 3)(7 — 8) + (5-38). 
10. (7+ 3)(7— 3) + 5-3. 
11. (138 —*6)(18 — 8+ 8 + 4). 
12. 88+ 4-10. 
13. (84 2)a+ 5(44 2)a. 
14, Qa + 3y + 3a + 4y. 
15. 2a+604+c+a-+ 204 Ze. 
16. 2 [7(2+ 6 — 4) + 244 32]. 
17. 2{(8+ 4) (6+1)4+ (24+83)}. 
18. 5 [(8+7)4+3 (14+2+8)]. 

16. Evaluation of expressions. It is frequently necessary 
to find the numerical value of an expression for certain values 
of the letters involved. This process is called the evaluation 
of the expression. Such evaluation will be used later to test 
the accuracy of the results of algebraic operations and to check 
the answers to certain problems. | ' 


| 


4 


‘Anrs. 16,17] EXPRESSIONS IN ONE LETTER Ay 


t 


EXERCISES 


In each of the following, find the value of the expression 
for a = 3, b = 4, c= 5, d = 2, and simplify the results: 


Peo. 9. ab+ be — ad. 
2. (a+ b)(c+ da). 10. 4ab — cd — d. 
3. (a+ b)(c — d). 11. BF — c+ a. 
4. 4a°b. 12. 3c? + 2a? — d. 
5. 2ab(c — d). 13. 2b+ (5c —d) + (a+b). 
6. 6ab— b—c. 14. 6b+ 10bc + 8a — d. 
7. (a + 6)? — 2(c — d)?. 15. 4ab? — cd?. 
8. 5c + (b— d) + 2(a+ Dd). 16. 5bc? — 16(5a — 30). 
| _-_—_ 


17. If Q represents the number of gallons of water flowing 
from a pipe per minute, v the rate of flow of the water in feet 
per minute, and d the diameter of the pipe in inches, it is known 
that 


Q = .04vd?. 


Find the number of gallons of water flowing per minute through 
a pipe 2 inches in diameter when the rate of flow of the water 
is 100 feet per minute ; through a three-inch pipe when the rate 
is 75. 


1%. Expressions containing one letter. Algebraic expres- | 
sions which contain only one letter form a class of great im- 
portance. It is desirable to note the change in the numerical 
value of the expression as a succession of numbers is substituted 
for the letter. 


Example. Show that the expression, 2? + 12—-— 7x, decreases as x 
Jakes on the succession of values, 0, 4, 1, 3, 2, 3, and 3. 


/ Souvurton: If z = 0, then x? + 12 — 7x equals 12. 
If x = 3, then x? + 12 — 7x equals 8}, and so on. 


Ve can show the change in the expression by a table as follows: 





18 ALGEBRAIC EXPRESSIONS [Cuap. II. 








x a+ 12 =e 
0 12 

} 8} 

1 6 

i 34 

2 2 

5 3 

2 q 

3 0 





eg this table we see that as x increases from 0 to 3, the expression, 
+ 12 — 7x, decreases from 12 to 0. 


EXERCISES 


1. Find the value of 2x +3forz=0; forz= 1;forg= ae 
fOr 5, 

2. Make a table showing the values of 5a — 3 when zx takes 
on the values, 1, 2, 3, 4, 5. 

3. Make a table showing the values of 7 — 3y for y = 0, 
2.4; .6,..8, anda: 

4, Show that 5z— 1 increases as z takes on the succession 
of values, 1, 3, 2, 8, 3, 3, and 4. : 

5. Show that 16 — 2z decreases as z takes on the same 
values as in Exercise 4. 

6. Make a table showing the eR of 22+2+1, for 
Ter ae, 4: 

7. Make a table showing the values of 2?+ 2 — 2z, for 
x = 1, 2, 3, 4, 5, 6, 7. Does the expression increase or 
Rerrenas . 

8. Make a table showing the values of 2?+ 2 — 2z, for 
x= 0, 1, 4, 3, 1. Does the expression increase or decrease for 
these values of x? 

9. If a is the length of one edge of a cube, and V is the 
volume of the cube, write in symbols the rule for finding the 
volume. From the result make a table showing the volumes of 
cubes with edges 4, 1, 13, 2, 25 and so on to 5. | 

10. If $100 is it at 5% simple interest for a number of 
years it amounts to 100 + 5n dollars if n is the number of years. 


; 





“Arts. 17,18] GRAPHICAL REPRESENTATION 19 


From this make a table showing how $100 increases every year 
up to 10 years. 

11. Show in symbols how $100 increases if cae at 6% 
simple interest. Tabulate the results as in Problem 10. 

12. If v represents the velocity of the wind, and P is the 
pressure of the wind on a pane of glass one foot square, then 
it is known that 

P= 107 + 225, 


‘Find the pressure when the wind is blowing at the rate of 15 
miles per hour ; 20, 30, 40, 50 miles per hour. 


18. Graphical representation. The way in which an ex- 
pression involving one letter changes “as the value assigned to 
‘the letter is changed, can be represented to the eye by a dia- 
gram. This is shown as follows for the expression 2x + 3. On 
the horizontal line of Fig. 3 mark the numbers 1, 2,3,4... 
Make a table of values for 2x +3. Thus we have: 
| Pij2i3] 4] 5] 

Sa K 17 | 9 | 11} 18] 15 








“At the point marked 1 draw a vertical line of length 27+ 3 
when x = 1; that is, of length 5. At the point marked 2 erect a 





0 I 2 3 4 5 6 
Fiaes 


‘vertical line of length 2x + 3 when xz = 2; that is, of length 7. 
Proceed in the same way for each value of x given in the table. 





a 
4 


20 ALGEBRAIC EXPRESSIONS [Cuap. IL. 


The diagram will then present to the eye the way in which 
27 + 3 increases as X increases. 

Any table of numbers, whether representing an algebraic 
expression or not may be represented to the eye in a like manner. 
For example: During the opening day of a new shoe store a 
record was kept of the number of customers for each hour of 
the ten hours the store was open. 


| 
| 








Hour | 
Number of customers | 


HH) bo 


11 
2 





This table is represented in Fig. 


10-7 
LD 
8 





E 7 
=. 6 
iS) 
3 5 
2 
g4 
a 

3 

2 

af 

i we 3 4 5 6 7 8 a 10 
Hours during which the Store is open. 
Fia. 4 


In these diagrams any convenient unit can be used for the 
lengths of the vertical lines. 


EXERCISES 


1. Show by a diagram the values of 5% — 3 when z takes on 
the integral values 1 to 6. 

Represent by a diagram the following expressions when x 
takes on the integral values 1 to 6. 

AE i Neb 4. 2P+a2+1, 

3. 3x — 1. 5. v4 2— 2a. 

6. The morning temperature record of a fever patient is 
given by the table: 


(Ber. 18] GRAPHICAL REPRESENTATION 21 


_ [Morning [1 415 16171819 
temperature 1.8 | 1 | 9 | 6 | 0 


3 | 

Ts 2 
Represent this table by a diagram. 
7. The weights of a baby weighing eight pounds at birth 
for each month of the year are given by the table: 
Month |1 |2 |3 
Weight | 93 





tbo 
| 





































Paes | eric s. ot 10 pli 
142 | 153] 16] 18] 19] 192 | 20] 21 | 22 








Represent by a diagram. 

8. The heights of the same baby for the same months are 
19 inches at birth, then 203, 21, 22, 23, 235, 24, 245, 25, 253, 
| 26, 263, 27 inches. ae By a ae 


EXERCISES AND PROBLEMS 


1. The horse power of a certain style of gasoline engine 
is given by the expression | 
H=<.DN} 
where Z is horse power, D the diameter of the cylinder in inches, 
and N the number of cylinders. Find the horse power of a two- 
cylinder engine with 5-inch cylinders; with 6-inch cylinders. 
2. Write in algebraic symbols: The volume of a sphere is 
four thirds the cube of the radius times 7. 
825. 27—- 3° + 3=? 4.273) — 18-5 Sia 
5. In the formula s = 16.12, ¢ represents the number of 
seconds a body has been falling, and s.represents the distance 
it has fallen. How far will a stone fall in 4 seconds? 
6. Give the formula of Problem 5 in words. 
7. Make a table showing the’ distance through which a 
stone will\fall in 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 seconds. 
Find the values of the letters in the following: 


Se oy=.25; - 10. 32 = &4. 
9. 4+ a= 19. Hea ot 27 =120: 


* It is not expected in this and similar problems that the teacher will 
take the time to explain the physical principles involved. 


22 ALGEBRAIC EXPRESSIONS [Cuap. IL. 


12. A has $550 and B $150. How much must A give to 
B in order to have just twice as much as B? 

13. If a is equal to 2 times b, express 6a + 36 in terms of 3 

14. Write 6 factors of 2a(a + b). 

15. 42? — 32°+ 5a?- w=? 

16. If x is 3 greater than y, express 7x + 2y first in terms 
of x and then in terms of y. 

17. Find the value of z{[(#+ 1)? y—1)]+ a when 
e=i2and ya 4. 

18. 7 (a— b) — 2(a— b) — 38(a — BD) = (2) - (a — DB). 

19. Write in symbols: The square of the sum of a and 6 
divided by their product. 

20. Find an expression for the surface of a cube if one edge 
is given. From the result make a table showing the surfaces 
of cubes with edges 1, 2, .. . to 6. 

21. A bag contains x white balls, three times as many black 
balls as white balls, and twice as many red balls as black balls. 
How many are there of each color if there are 40 balls altogether? 

22. Given that 254 centimeters equals 100 inches very 
nearly. Using c for number of centimeters and 7 for number 
of inches, show in symbols the relation between inches and 
centimeters. Tabulate the result up to 12 inches. 

23. Show by a diagram the temperature record of a fever 
patient. The degrees of fever recorded were 2, 5.4, 4, 6.1, 4.5, 
6.5, 5.3, 6.7, 4.5,.6.5, 5.9, 6.2, 7, 5, 6.7, 5.8, 6.107 

24. The following table gives the length af a child’s. bare 
foot when the size of the shoe is known. 











ae of shoe 








Show by a diagram the relation between size of shoe and length 
of foot for children. 


CHAPTER III 


POSITIVE AND NEGATIVE NUMBERS 


19. The use of a scale to represent the numbers of arith- 
‘metic. The numbers of arithmetic may be represented by 
points on a straight line. 

Let OX be such a line (Fig. 5). Adopt a unit of measure, 
AB. Begin at O and divide OX into intervals of length AB. 
‘We may thus obtain a scale of indefinite length. If the ends 
of the intervals are marked 0, 1, 2, 3, 4, . . . as in the figure, 


sei. oss 45> 6. y.8 8 


Fiq. 5 


we have a point on the number scale corresponding to each of 
these numbers. 

Fractions may also be represented by points on this line. 
Thus, there corresponds to. the number 4 a point midway 
between the points marked 0 and 1; to 33a point 4 of the dis- 
tance from 3 to 4; and, in the same way, for every fraction 


there corresponds a point on the scale.’ 


20. Addition and subtraction on the scale. Addition and 
subtraction may be performed on the number scale. 

Thus, to add 3 to 4, begin at 4 and count 3 spaces to the 
right. To subtract 3 from 4, begin at 4 and count 3 spaces to 
the left. ; 

In general, to add a to b, begin at b and count a spaces to 
the right ; and to subtract a from b, begin at 6 and count a 


spaces to the left. 
23 


24 POSITIVE AND NEGATIVE NUMBERS [Caap. IIL 


EXERCISES 

1. What number is represented by each of the following 
points? 

(a) The point 3 spaces to the right of 5. 

(b) The point 4 spaces to the left of 11. 

(c) The point midway between 4 and 5. 

(d) The point midway between 33 and 4. 

(e) The point one-third of the distance from 2 to 3. 


2. State in words the positions of the points that represent 
the numbers 3, 4, %, %, 451, 0. 
3.-On an ordinary ruler perform the following additions 


and subtractions: 


(a) 542. (e) 24+3-4. (h) 23+ 1. 
(b) 5-2. (f).7-2-3. (4) 8—@= 4; 
(c) 5—5. (9) $+2 (7) Ga ae} 
(dq) 24+344. 


21. Positive and negative numbers. If, on the scale in 
Fig. 5, we attempt to subtract a number from a smaller 
number, say 6 from 4, we get off the scale. 

Let the scale be extended to the left of 0, Fig. 6. If we now 
attempt to subtract 6 from 4 by counting 6 units to the left of 
4, we arrive at a point 2 units to the left of 0. 

Awe 3 

-11-10-9 -8 -7 -6 -5 -4-3-2-1 012 3 45 67 8910 

Fia. 6 

The numbers that correspond to the points to the left of 0 
are called negative numbers. 

The numbers of arithmetic, which are represented by points 
to the right of 0, are called positive numbers. 

The minus sign is used to distinguish the negative from the 
positive numbers. 

On this scale then, 

4—6= -2. 


Ants. 21, 22, 23,24] POSITIVE AND NEGATIVE NUMBERS 25 
Similarly, 
: 7-—10= —8, and 2—- 8= -6. 


The essential thing in the above representation of positive 
and negative numbers is that they are marked off in opposite 
directions. 

In Art. 1, we observe that the fraction is introduced to make 
division possible when the quotient is not an integer. In a 


similar manner, the negative number is here introduced to make 
_it possible to subtract from a number a greater number. 


22. Illustrations. The student has had experience in arith- 
‘metic with quantities measured in opposite directions. 


' The thermometer gives a simple example. Temperatures 


' 


above and below zero are measured in opposite directions. It 
is simple and convenient to regard temperatures above zero as 
positive and those below zero as negative. 

If the temperature is 10° above zero, or +10°, and falls 
30°, we say it is then 20° below zero, or —20°. This result is 
obtained algebraically by saying 10° — 30° = —20°. 

North latitude is commonly called positive and south 
latitude negative.. If a ship is in latitude +20° and sails 30° 
south, it is then in latitude —10°. Algebraically, 


20° — 30° = —10°. 


Positive and negative numbers are often called \signed 
numbers. \ ; 


+238. Numerical or absolute value. The numerical or ab- 
solute value of a number is its value without regard to sign. 

The absolute values of —10, +380, +6, —9, are 10, 30, 
6, and 9, respectively. 


24. Greater and less. The expressions greater than and 
less than which are common to every day life, when used in the 
precise sense of algebra, are easily misunderstood. For this 
reason, we point out their meanings on the number scale. A 


26 POSITIVE AND NEGATIVE NUMBERS [Caap. III 


number a is greater than a number b when a is represented by 
a point on the scale to the right of that representing b. 

Thus, 1 is greater than —4; a temperature of —10° is greater 
(or higher) than one of —20°. 


Fia. 7 


Exercise. Explain by the use of points on the number scale what is 
meant by the expression ‘‘a is less than b.” 


EXERCISES AND PROBLEMS 


1. Locate on the number scale: —5, —2, 0, —8, —%, 
+5. 

2. Find the point that corresponds to the following dif- 
ferences: 5—11; 7-7; 4-45; 6-5; 0-4. 

3. Using temperature above zero, north latitude, west 
longitude, and assets as positive quantities, represent the 
following as signed numbers: 12° above zero; $10 debts ; 
30° west longitude ; 40° south latitude ; $100 assets ; 1° below 
zero. 

4. A goes 6 miles west, then 18 miles east, then 5 miles 
west, then 2 miles east. Using distarice east as positive, write 
the above distances as signed numbers. How far and in what 
direction is he from the starting point? Show how to find the 
answer by using the number scale. 

5. If the temperature is +10°, falls 13°, and fen rises 
5°, what is the resulting temperature? 

6. The temperature is now 60°. What will it be after it 
falls: .(a) 20°; (6) 60°; (c) 70°? 

7. If the temperature is —10°, what will it be after it rises: 
(2) 529,(b) 102 ae) 20g 

8. Give the absolute values of +8, -—10, -—%, +443, 
—8.5. 

9. What number is one greater than —8? What number 
is one less? 


| 


“Ant. 24] EXERCISES AND PROBLEMS 27 





- 10. Show on the number scale the distance between (a) 
“+7 and +4; (b) —7 and —4; (c) —7and +4; (d) +7 and —4. 
11: Ina game, A loses 10 points, gains 7, gains 31, loses 14, 
loses 36, gains 5. What is his final score? 

12. A has $375 assets and no debts ; B has $200 debts and 
no assets. Consider debts as negative assets. What is the 
sum of their assets? How much more assets has A than B? 

13. What should be taken negative if the following are con- 
‘sidered positive? 


(a) West longitude. (f) Cubic inches of expan- 
(b) Dollars gain. sion of a toy ballon. 
(c) Points by which a game (g) Excess of water pumped 

is won. from a well over that 
(d) Miles northeast. flowing. into it in the 
(e) Excess of persons enter- same time. 


ing a room over those 
leaving the room. 


14. What would be the meaning, if any, of a negative result 
in finding — 
(a) How much money you had gained in a trade? 
(b) The score by which your football team won the last 
- game of the season? 
(c) How much more a cubic foot of a certain liquid weighs 
than a cubic foot of water? 
(d) The rate at which a lion ran away from a hunter? 
(e) The increase of the population of a town in 10 years? 
(f) How far Philadelphia is west of New York? 


Historical note on negative numbers. The extension of the concep- 
tion of number to include the negative was an exceedingly slow process 
in the development of algebra. The Hindus recognized the exist- 
ence of the negative in working at the quadratic equation perhaps as early 
as 800 a.p., but they had little to say about such numbers except that 
the people did not approve of them. It was not until the work of Des- 
cartes (1596-1650) (see p. 182) that the rules for operation with negative 
numbers were at all well understood. 


28 POSITIVE AND NEGATIVE NUMBERS [Cuap. IL 


(g) The height of the bottom of a well above sea-level? 
(h) The age of Washington at the time of the surrender 
of Yorktown? | 


25. Addition of signed numbers. ‘Thus far it has not been 
indicated what is meant by the sum of two numbers when one or 
both are negative. Some illustrations will suggest how such 
sums should be defined. | 


Example. How many steps and in what direction from the starting 
point is a person who takes — 
(a) 8 steps east and then 5 steps east? 
(b) 8 steps west and then 5 steps west? 
(c) 8 steps east and then 5 steps west? 
(d) 8 steps west and then 5 steps east? 
So.tuTion: Let steps east be considered positive and steps west nega- 
tive. The answers may be obtained by counting on a scale. 
(a) To add +5 to +8 we begin at +8 and count 5 units to the right, 
arriving at +13. 
(b) To add —5 to —8 we begin at —8 and count 5 units to the left, 
arriving at —13. 
(c) To add —5 to +8 we begin at +8 and count 5 units to the left, 
arriving at +3. 
(d) To add +5 to —8 we begin at —8 and count 5 units to the right, 
arriving at —3. 
Hence to add a positive number we count to the right, and to add a nega- 
tive number we count to the left. 
The results just obtained may be stated as follows: 
+8 + (+5) = +18, 
Sid 8 eee sie 
+8 +(-5) = 43, 
—-8 + (+45) = -3. 


- The foregoing discussion suggests the following definitions 
of sum when signed numbers are added : 
The sum of numbers with like signs is the sum of their absolute 
values preceded by their common sign. 
The sum of numbers with unlike signs is the difference of 
their absolute values preceded by the sign of the one having the. 
greater absolute value. 


(ARTs. 25, 26 | il OF SIGNED NUMBERS 29 
EXERCISES 
Perform the following additions by counting on a scale: * 
‘me ) 10 ee G27 13) as ie oS 
t o 12 4 ~9 
on 2 4 
2;..—10 Ge.eh —6 1 2 
ee! ek s i wae 
3. —10 7. —2 —§ 14. 9 16 10 
7 ly. = ~6 —12 
—3 4 
4, 10 8. 4 12. -—9 4 6 
= =5 4 a a 
Answer the following questions : 
ast f= 13. 22. —4=-13+7 
18.-?+8= 6. 23. ?+6=0. 
Ieee) = 7. 24. —124?=0. 
omen? — 12. $5. 3=-7+? 


Civetew = —10. 


Find the value of x in the following : 


26. (7+ 8 = 12. 31. -—-l-—2z=-3. 
fe i= 3. 32. 20+2z=0. 
28. x-—11=-8. 33. 13-—2=0. 
29. 4-—2z=-16. 84. 2+(-—11) =15. 
30, 1b= xz — 9. 35. 2-2 = —2, 


26. Subtraction of signed numbers. As in arithmetic, 
subtraction is the process of finding one of two numbers when 
their sum and the other number are given. 


6 because 6+4 = 
-14 because -14+4 = —10. 
14 because 14+ (-4) = 
— 6 because -6+(-—4) = -10. 


Thus, +10 —- (+ 4) 
Pigeit+ 4) 
+10 — ( -4) 
10= (4) 


30 POSITIVE AND NEGATIVE NUMBERS [Cauap. III. 


EXERCISES 


Fill in the blanks in the following : 
1. 47 = (+44) 2) eines (0) aaa 
2. —7—(-—4) =(), since (—4)+ ( ) = -7. 
3. +4-— (+7) = ( ), since (+7) + ( ) = +4. 
4. +7— (-4) = (), since (—4)+( ) = +47. 


In a similar way perform the following subtractions : 


& 3-— (42). 13. 8 — (+48). 

6. —3— (42). 14. 7-— (+49). 

7 —3— (—2). 15. O-—(-—?7). 

8 §—(—5d): 16. O- (47). 

9, —4—(—4), 17. 27-—(-9). 
10. -—9— (10). 18. —8—(+18). 
11.-.—] — (+2): 19. 30-— (-8). 
12, —1—(-lI). 20. —8— (+30). 

Answer the following questions : 
21. ABs 25. 8—- ?=<-5. 
22. ?—6=0. 26. —7= fas 
O35 7 LP = tae 27. —-5-—14=? 
24, -—8—2=4. 28. —10— ?=5. 


2%. Subtraction on a scale. We have seen, Art. 20, that 
to subtract a positive number on the scale we count to the left. 
Thus, in Art. 26, the result in Exercise 1 can be found by 
beginning at +7 and counting 4 spaces to the left, arriving at 
+3; and the result in Exercise 3 can be found by beginning 
at +4 and counting 7 spaces to the left, arriving at —3. 

We can also subtract a negative number by counting on the 
scale. The answer to Exercise 2, Art. 26, can be found by 
beginning at —7 and counting 4 spaces to the right, arriving at 
—3,; and the result in Exercise 4 can be found by beginning — 
at +7 and counting 4 spaces to the right, arriving at +11. 


Arc. 28 | _ RULE FOR SUBTRACTION ol 


_ We have then the principle: To subtract a negative 
number, count to the right the number of spaces indicated by 
the number. 


Exercise. Find the answers to Exercises 5-20 in Art. 26 by counting 
on the scale. 


28. Rule for subtraction. Any problem in subtraction 
may be changed to a problem in addition. 


Examples. —© 

+10 — (+5) = +10+(-5) = +. 5. 
=10\— (+5) = -104+ (45) = — 5. 
Sere nyt 10 (5) =. ~ 15. 
+10 — (—5) = 410+ (+5) = +15. 


ry 4+ 1 Od 
We then have the working rule: To subtract any number 
change its sign and add the resulting number. 
EXERCISES 


Perform the following additions : 


al an 1g Toten Ba hacanl 
» eit ~9 2 3 
EF 3 sly ai 
"eal Rad 62> .,-9 ) os 
~8 pals pe 101 
= ree 2: 7 
3. —11 Goyal ¢ 6 4 
ot eld -3 —21 


Perform the following subtractions : 
ao —. 1. 14. 9 — 17. 17. —3 —(-14). 


seat): 16,9 — (—17).. 18. TT —/18. 
ots 7-8. TGA Dita hdd ata 19 pers Bh 3 (224), 


o2 POSITIVE AND NEGATIVE NUMBERS [Cuap. IIL. 


Answer the following questions : 


Sos See a7. -7— (2) 2 or 

R183) aoe foe yee ) 

pie Be 29, —17.— (1 

GSO yess ae 1) 


4. 1-24+34+?7?=4 $81. 104 2s 
25. —21+11-—?=14. 32. ?+(-7) —(-7)4+7 =—7. 
26. —-3 —? = (-7) =0. 83. —.14 .5—- (-.6) —-Q@ =l1. 


EXERCISES AND PROBLEMS 

1. From the sum of —2$, —381, and 123, subtract the sum 
of 3, 5, and 84. 

2. What is the difference between the temperatures of 
+85° and +40°? Of +36° and +14°? Of +438° and - 209 
Of —43° and —51°? Of —24° and —6°? 

3. If a man is worth $20,000, how much must ie lose to 
be in debt $8400? 

4. B has assets of $300, $100, $800, and $1000, and debts 
of $200, $400, $100, and $600. Let assets be represented by 
positive numbers and debts by negative numbers. What is 
the net value of B’s property? What would be the net value 
of his property if (a) the courts should cancel the debts of 
$300 and $100; (6) he should lose $400 worth of property by 
fire; (c) he should gain $400 in a trade? 

5. Suppose that weights to the amount of 300 pounds are 
attached to a balloon which pulls upward with a force of 400 
pounds. Let pull upward be regarded as positive and pull 
downward (weight) as negative. What is the net upward 
pull? What would be the net upward or downward pull 
if (a) 200 pounds of weights were removed; (6) gas with 
a lifting force of 200 pounds were added; (c) 200 pounds of 
weights were added; (d) gas with a lifting force of 200 pounds 
were removed? 

6. In the difference a — 6 = 6, the value of a runs through, 
the series of whole numbers from 10 to 0. Find the correspond- 


Anr. 28] EXERCISES AND PROBLEMS 33 


ing values of 6. Make a table showing these corresponding 
values of a and b. 

7. In the difference x — y = —4, the value of y runs 
through the series of whole numbers from —5 to +5. Find the 
corresponding values of x, and make a table as in Exercise 6. 

In the following problems + denotes A.D., and — denotes 
B.C. 

8. The oldest mathematical manuscript known was writ- 
ten by an Egyptian named Ahmes. The date of the manu- 
script is thought to be about —1700. How old is it? 

9. The Greek geometer, Euclid, lived about —300. How 
Jong was that before the birth of the French geometer, Des- 
eartes, who was born in +1596? 
~ 10. Some characters from which our present numerals are 
thought to have developed, are found in inscriptions made in 
India as early as —250. The first undoubted use of the zero 
in India is said to have been in the year +876. How many 
years between these dates? 

11. Archimedes was born in the year —287, and died in 
the year —212. How old was he when he died? 

12. Through how many degrees of latitude does a ship sail 
in going from latitude —21° to latitude —56°? 

13. Through how many degrees of longitude does a ship 
sail in going from longitude —17° to longitude +35°? 

State what single change will produce the same result as the 
changes mentioned in each of the following exercises. 

14. The temperature rises 20°, falls 13°, then rises 6°. 

15. An elevator goes up 60 feet, goes down 24 feet, goes up © 
48 feet, and then goes down 84 feet. 
16. A traveler goes 350 miles east, and then 420 miles 
west. 

17. In a race a runner gains 12 feet on his opponent, loses 
7 feet, gains 2 feet, and loses 9 feet. 
18. A political party gains 724 votes, loses 328, loses 35, 
and gains 120. 





34 POSITIVE AND NEGATIVE NUMBERS [Cuar. IL 


29. Addition and subtraction of several numbers. In add- 
ing and subtracting several numbers we may proceed from left 
to right performing each addition and subtraction as we come 
to it. For example, ¥ 


+3 + (—2)—3-— (-1) = 7+ (—2) 3 Sia 


5. Sean 
-2—-(€9) 
at 


It is usually shorter, however, to remove the parentheses, 
unite the positive terms and the negative terms, and then com- 
bine these results. 

Thus, 4+3 + (-2) —-3 —(-1) =443 —-2-34] 
=8-5 
sro. 

The second method is usually more convenient when the 

numbers are written in columns. . 


EXERCISES 


Perform the following additions and subtractions: 
1.62—-3+4+546-7-48. 

2. —2 — (-3) + (-4) — (—5) + (-6) - (-7). 

3. 1 + (—2) — 34 (—4) - (5). 

4. 2a — 3a +a — 4a + 6a. 

5. 54 — 64 + 2 — Tx + 2x — (42). 

6. —1-— 2- 3- (-4) — (—5) — (-6). 


7 —8 9. —59 11. —5m 13. ory a 
+7 +21 +9m —4.6xy 
—3 +22 —4m —.lry 
—11 Bak: +7m 5.22y 

8. —17 10. +33a? 12. —45y 14. ee 
—25 —18a? —I13y 2a 
+40 —21a? —20y — —2a 


+43 +6a? +50y —a 


a 
; 
"Ants. 29, 30,31] MULTIPLICATION IN ARITHMETIC 30 
aie 


15. 8.5 18. 3(2)? 21.. $2 24. —Txry 
—.75 5(2)? —2Qx gry 
_ 05 _(2/ _ 4a —rsty 
16. —3.00 19. 4a 22. Ray? 2028 
42 —8a — ery? P: 
—.01 _ 2a SR ex a 
| $3 
a 
G 
7.3 20. Sab 23, 4m 26. «+57 
—4-3 —4Aab am —2.7 
2:3 —ab —im —9.7 








30. Multiplication in arithmetic. So long as the multiplier 
is a positive integer, multiplication may be defined as the pro- 
cess of finding the sum of a number of equal numbers by the 
‘use of the multiplication table. 

The result in a multiplication is called the product. 

This definition of multiplication has no meaning when the 
multiplier is a fraction. Thus, to say that the product, # x 2, 
means finding the sum of ? equal numbers has no meaning. 

Hence, an extended definition of multiplication must be 
made when the multiplier is a fraction, and we say that the 
product of two fractions is the product of their numerators 
divided by the product of their denominators. 

31. Multiplication of signed numbers. It is necessary 
again to extend the definition of multiplication in order that 
the products of positive and negative numbers shall have a 
meaning. 

For example, we are to define what is meant by such, pro- 
ducts as 3- —4, -—3-4, and -—3- —4. 

We have said that 3-4 =4+4+4=12. Using the same 
meaning of times, we say that 3 - —4 = (—4) + (-4) + (-4) = -12, 


36 POSITIVE AND NEGATIVE NUMBERS [Cauap. IIL 


which extends multiplication to the case of multiplying a 
negative number by a positive number. 

Since to multiply by +3 we add, it seems reasonable to say 
that to multiply by —3 we subtract ; and we say that —3°4 
means that 4 is to be subtracted 3 times. That is, 


—3:4= -4-4-4 = -12. 


Similarly, —3-—4 means that —4 is to be subtracted 3 times. 
That is, 
=3,-4=-(-) — Cj 
= Ach du Ape 


In general symbols this discussion may be summed up as 
follows : 


OME hopcefs))° 
a- —b = —ab, 
—a-b = —ab, 
—a-—b = ab. 


We may then state in words the 

Rule of signs for multiplication. The product of two num- 
bers with like signs is positive, and the product of two numbers 
with unlike signs is negative. 

It follows from the law of signs that a product is positive if 
it contains an even number of negative factors, and is-negative 
if it contains an odd number of negative factors. Thus, 


—2:3.-1-—5 ene 
and -1.4:-2+ 8.6. 22. 


EXERCISES AND PROBLEMS 


Perform the following indicated operations : 


1. 4- —5. 5. 4-—2. 

2. —4-5. 6. —2-—-a. 
3.. —1 - —2. 7. 8-0. 

4. —7-—-6. 8. —2--3-4. 


Arr. 31] EXERCISES AND PROBLEMS 37 


9. 4--5:-6 17. 8--9--5-2. 

10. 3-0: —2. 18.03 '°457 52 Oa 
et - 7: —3-2 19. (—2)?. 

12. —2--a:7z. 20. (—1)°. 

mo 2-345: —4 21. (—3)*. 

14. 3-104 (-3- -8) 22. (—2)?- —d. 

15. —a-b-x-m x 93.0 (1) - (2)? (23)F. 
16. —-2--3-4-2 24. (—a)*. 


m 25. -15 = —5-3 =5- —3. Find two pairs of factors of 
—21; 14; 95; -33; —ab; xy. 

26. If y = 2x — 4, find the values of y when «x takes on the 
‘integral values from —2 to 6. Represent the values of y by a 
diagram. 

SoLUTION: 
Pee oe iO Ly 2 | 8:)-4 1.5 | 6 
-8 | -6 |-4 |-2|0[2/4/6|8 
The corresponding values of x and y are given in the table. The 
diagram (Fig. 8) is made as in Art. 18. Notice that when the values of 
y are negative the lines representing those values extend downward. 














Fic. 8 


27. If s = @, find the values of s when ¢ takes on the integral 
values from —4 to 4. Make a table showing the corresponding 
values of s and ¢, and represent these values in a diagram as in 
Exercise 26. 

' 98 The formula C = 3(F — 82), gives the temperature, 
C, on a centigrade thermometer in terms of the temperature, 





38 POSITIVE AND NEGATIVE NUMBERS [Cuap. III, 


F, on a Fahrenheit thermometer. Find C when F has the val- 
ues —20°, —10°, —5°, 0°, 5°, 10°, 28°. Make a diagram repre- 
senting these values of C and F. 

29. If y = 2 — 62, make a table of the corresponding values 
of « and y, when x takes on the integral values from —2 to 8. 
For which of these values of x does y have the least value? The 
greatest? For what values of x is y equal to zero? Between 
what values of « does y increase? Decrease? 


32. Division of signed‘numbers. Division is the process 
of finding one of two factors when their product and the other 
factor are given. 


Thus, 20 + 4 = 5, since 5° 4 = 20. 


The given product is called the dividend, the given factor 
the divisor, and the factor to be found the quotient. 


The application of this definition gives results as follows: 


+14 * F 
ra +7, because +7-+2 = 414; 
—14 

=n +7, because +7--2 = —14; 
is = —7, because —7--2 = +14; 
—14 

aes —7, because —7-+2 = —14. 


We have then the 

Rule of signs for division. The quotient of two numbers 
with like signs is positive, and the quotient of two numbers with 
unlike signs is negative. | 

In algebraic symbols, this rule is, 


+a a 
Fees; 
Bn 
=} aeagne 


‘Arr 33] . FRACTIONS 39 


sete ees 
aj Ba Me 
de 
Sse Y, 


33. Fractions. A fraction is an indicated division. Thus, 
#means3 +5. The terms of this fraction are 3 and 5, 3 being 
the numerator, and 5 the denominator. 

In arithmetic we had the principle: 

The numerator and the denominator of a fraction may be 
multiplied or divided by the same number without changing the 
value of the fraction. 

This principle holds for the numbers of algebra, and is of 
great service in simplifying fractions. 


EXERCISES 


Perform the indicated divisions and check the results by 
multiplication. 





ee Eee 0d Rosie ad tk 
| 19° nee ty, A. 
eee ay 2 Ra 8 2 
eee 1) SO. Sesnh = G. a Bes ot a 
14. wals 18. AY 
meine 2 10.<—ab ~ a. —3 —2--3 
oy ae 
ley ae be ioe Be Sg 
—¢ —* 2 
. >i ere ‘Fe ahaa bs OY, ata 
—2X xy 2 
Simplify the fractions : | 
a 4a = 102 
ahs ates ys 12° 25. os ; 
=8-5 122 ae 
22. Sans. 24. re 26. ab? 


40 POSITIVE AND NEGATIVE NUMBERS [Cuap. IIL 











E 1 4 
cy peasnasie 30. 4: Pegged 
vo = 4a 
Freee te ai ee sae 
2 v DI 
8 7x 3a+7 
29. 32. or ae 


36. What number divided by 2 equals 8? 
37. Find 2 if 5 = 8. 
38. What number divided by —3 equals 4.5? 


39. Find 2 if —- 4.5. 


In the following exercises, the letters stand for unknown 
numbers. Find the numbers. 


x 6 48 
40. 6 = 2. 45. 3a = 7 50. ye = 6. 
x 42 
41. 7, - 2: 46. eS S68! Bi. = = -3. 
Lo HET as 7 ee 
42. 5 = 5 47. = -5 52 3 = —0.08 
Age ee iGo ee 
5 8 
44, = 5. 49. : eS 


MISCELLANEOUS QUESTIONS AND EXERCISES 


1. How much greater is 9than 4? 9than —4? 52 than 
-2x? athan 7? 

2. Write in the form of a fraction the quotient of 17 di- 
vided by 20. Also the quotient of a + b divided by m; of the 
sum of the cubes of a and 6 divided by the sum of a and 6; of. 
n divided by a number 6 less than 5 times n. 


J 


‘Art. 33] MISCELLANEOUS QUESTIONS 4] 


3. What is the meaning of a®? Of 6a? Find the differ- 
ence between a® and 6a when a has the values —2, 0, 2, and 3. 

4. Write the following using exponents: 2-7-7-7-7; 
S-m-n-n-mM-mM; L-¥- yz LY 2-2-2 8-4-5 -4-5-3-3, 

5. How is the sum of numbers with like signs found? Of 
numbers with unlike signs? 

6. When is the remainder in subtraction greater than the 
minuend? | 

7. Perform the following subtractions on a number scale : 
8-9; -4-7; 8-—(-8); -5-(-6); -—7—-(-12). 

8. What is the definition of multiplication when the mul- 
tiplier is a positive integer? To what other cases have we ex- 
tended the meaning of multiplication? 

9. Give the rule of signs for multiplication. 

10. Find the following products by adding or subtracting : 
Mee 4- 4-9; —5- —8. 

- 11. What is the absolute value of .—45; 34; —34; .07; —3? 

12. Define division. State the rule of signs for division. 

13. In 6abz, state the coefficient of abx; of x; of ab; of 6a; 
of b. 

14. Give two illustrations showing how algebraic symbols 
may be used to shorten the statement of certain rules of 
arithmetic. 

3 

ee A af = an a find the corresponding values of y when 
x takes on the values —3, —2, —1, 0, 1, 2, 3. . 

16. If xy =1, find the corresponding values of y when x 
takes on the values —2, —1, —.1, —.01, —.001. Are x and y 
increasing or decreasing? 


CHAPTER IV 


EQUALITIES 


34. Members ofan equality. A statement that one expres- 
sion is equal to another expression is called an equality. 

The two expressions are called the members of the equality. 
Thus, in 5+ 38% =42, 5+ 32 is said to be the left-hand 
member and 42 the right-hand member of the equality. 

39. Identities. There are many different ways of writing 
the same number. Thus, 5 may be expressed as 7 — 2, 8 — 3, 
or 2+ 2 +1, and in many other ways. 

Likewise, an algebraic expression may be changed in 
form and still represent the same number. Thus, 5% — 4 and 
2% + 3x — 4 represent the same number no matter what num-— 
ber is represented by x. 

An equality in which the members merely represent dif- 
ferent ways of writing the same number is called an identity. 
Thus, the equalities, : 

5a — 4 = 274 324 -— 4 (1) 
and | 
x+ 2% = 32 maa 
are identities. 

Since the members of an identity are simply different forms 
of writing a number, the members are equal for all values of 
the letters involved. For example, the members in (1) and 
(2) are equal no matter what number we substitute for x. Try 
moet 253. andl, 


EXERCISES 


13 Show the 28 ia ae when a = 1, a = 5, a=0, 


and a = 10. 
42 


‘Ant. 36] . EQUATIONS 43 


2. Show that 3a + 2b =a+a+a+b-— 2b + 3b, if 
@= le -and 6 = 2: 
@=5 andb = 4; 
a = —1 and b = —2. 


=2 +2, whens =0,2=1,% =3,7=05, 





8, Show that ~—< 
x— 2 
and « = 10. 

36. Equations. An equality in which the members are 
equal only for particular values of the letters involved is called 
an equation. 

Thus, the equality x — 1 = 2 is an equation; for, the mem- 
bers are equal only when z = 3. 

EXERCISES 
1. Show that 7x —4=10 when x = 2, but not when 
@=1,orz = 3, orz = 4. | 
2. Show that 27 + 4 = 3% + 1 when 2 = 8, but not when 


ae 2. 
3. Show that 4% + 6 = x + 3 when x = —1, but not when 
— 1. 


Which of the following equalities are identities and which 
are equations : 
4.2% =%2. 
5B. oa + 42% = 7x42 -1. 
Hint: Try x = 2, and z = 1. 


2 

6. ao 
7 

7 22 =A. 


8. 3a4+1=27+7+1. 
9. 2? + 27 = -1. 
Hint: Try x=0. | 
10. 242% =27?+ 34-72. 
a. 2° = 4. 
fee — or + 2 = 0. 


44 EQUALITIES [Cuap. IV. 


13. 4 SS, 
14. 3a 4+ 2y = 3y —y+ 24442. 


3%. Solution of equations. In an equation, a letter whose 
value is to be found is called the unknown letter or simply the 
unknown. 

To solve an equation is to find values of the unknown which 
when substituted will reduce the equation to an identity. 

Such a value of the unknown is said to be a solution or root 
of the equation. 

When an equation is thus reduced to an identity, it is said 
to be satisfied. Thus, 1 is the solution of x +1 = 2; for, if 
1 is substituted for x, x + 1 = 2 is satisfied. 


EXERCISES 
Show that 2 is a solution of 2x7 — 3 = 1. 


Show that 3 is a solution of 4% + 1 = 13. 
Show that 0 is a solution of 4x + 2a = 7z. 


Show that — : is a solution of 7z + 9 = 5. 


foe 
Show that 2 and 3 are solutions of z? — 5a + 7 = 1. 


hehe Lact Cen 


Solve the following equations : 


6. Pe 8. +—4=2. 
7. 8+2=4. 9. ++1=5. 


38. Principles used in solving equations. The value of the 
unknown in an equation is unchanged by the following : 


(1) Adding the same number to both members. 
Thus, if 2 be added to each member of 2 + 3 = 5, we have 
-~@+5=7. Both equations have one and the same solution 2. 
(2) Subtracting the same number from both members. 
Thus, if 2 be subtracted from each member of x + 3 = 5, 
we have x + 1 = 38, and both equations have 2 for the value of : 
the unknown. 


ED 


“Ane. 38] VERIFICATION BY SUBSTITUTION 45 


(3) Multiplying both members by the same number other than 
ero. 

Thus, if we multiply both members of « +3 = 5 by 2, we 
| have 2x + 6 = 10, and 2 is the value of the unknown in both 
equations. The necessity of excluding zero as a multiplier may 
‘be seen by multiplying by x each member of 





+3 =5. (1) 
‘This gives 2 + 8x = 5a. (2) 


When zx takes on the value 0, equation (2) is satisfied, but 0 is 
‘not a solution of (1). 

(4) Dividing both members by the same number other than 
zero. 

| Thus, if we divide both members of 2x + 6 = 10 by 2, we 
have x + 3 = 5, and 2 is the value of the unknown in both 
equations. 





Division by zero is excluded from algebra. ° Thus ‘ has no 


‘meaning, and we should avoid attempting to divide both mem- 
‘bers of an equation by an expression that is zero. 


To illustrate briefly the use of the above operations, solve the equa- 
tion 5z — 9 = 6. 


SoLUTION: 

5r —9 = 6. 
Add 9, 5r -9+9 =6+9. 
Collect, be = 15, 
Divide by 5, f=3. 


39. Verification of solutions by substitution. The above 
operations (1), (2), (3), (4), Art. 38, are useful in finding values 
of unknowns, but the solution of an equation is not complete 
until the value of the unknown found has been substituted in 
the equation to test the result. This is called checking by 
substitution or verifying the result. 





¥ 


46 EQUALITIES [Cuap. IV. 
Example. Solve the equation 32+ 10 = 28. (1) 
SOLUTION: az +10 = 28. \ (2) 

Subtract 10, 3x + 10 — 10 = 28 — 10. MG (3) 

Collect terms, 32 = 18. 8 

Divide by 3, x = 6. 

CHECK: 3°6+10 = 28, 

28 = 28. 
EXERCISES 

Solve the following equations and check the results: 
sR ae Ps ees 11. 9y = 45 + 4y. 
Te ets en 12. 1l3y = —dy + 36. 
3.°2.4+3 = —Dd. 13. 54 +4 = —22+4 10. 
4. x—3 = —5. 14. 5n —4 = 3n +18. 
5. 7+4=9. 15. 8 — 62 + 12 + 82 = 24. 
6-2 —'3:=' 12. 16. 372 +5422 —1 = 24. 
7% 227+5=2-4. 17. +27 +10+2+422+10 = 140. 
8 31+7=27-11. 18 32-3442 —- 16 = 68. 
9. 524 +6 = 2x — 5. 19. 27 + 7 — 32x = 10. 
10, 32 +.3 = 22 —b: 20. 3yi2=y+8. 


40. Transposition. By the use of the principles (1) and 
(2), Art. 38, a number may be transposed from one member of 
an equality to the other by changing its sign. 


Thus, in the example, Art. 39, in deriving equation (3) from (1), 10 
may be subtracted from both members by omitting the 10 in the left 
member and entering a —10 in the right member. Likewise, to solve 


z-—5=7, 
we transpose —5, and obtain 
s=1(+52 12; 
After a little practice, this process of transposition of num- 
bers will sometimes be used to advantage instead of principles 
(1) and (2), Art. 38. However, the term “transposing” is not 


very essential as the process is simply that of subtracting the 
number from each member. , 


Arr. 41] TRANSLATION OF ENGLISH INTO ALGEBRA 47 


* 


Example. Solve the equation 5r+4 = 14. 


~ SoLurion: 5a +4 = 14. abs 
‘Transposing 4, we have 5a = 10. (2) 
. Dividing by 5, x =2. (3) 
CHECK: §°2+4=14, 
14 = 14. 


41. Translation of English expressions into algebraic ex- 
pressions. In order to give algebraic solutions of problems 
‘stated in words, it is necessary to develop skill in the transla- 
‘tion of English expressions into algebraic expressions. 


To illustrate, we give in parallel columns a few equivalent English 
‘and algebraic expressions, together with some statements of equality. 
Let s denote the length of a side, p the perimeter and a the area of a 
‘square. 





_ English expressions or statements Algebraic expressions or statements 
1. Four times the length of a side 1. 10 + 4s. 
) added to ten. 
(2. The square of a side plus 2. 2. s?42. 
8. The perimeter of a square equals 3. p=A4s. 
four times a side. 
4, The area of a square is equal to 4, a=s, 
| the square of a side. 
5. The perimeter of a certain square 5. 4s = s?. (See 3.) 
| is equal to its area. 
) 6. The perimeter of a certain square 6. 4s = 4s?. 


is equal to four times its area. | 


EXERCISES 


1. A rectangle is x feet wide and / feet long. It is twice 
as long as it is wide. State this last sentence in algebraic sym- 
bols. What represents the perimeter? 

| 2. If the sum of two numbers is 8, and one of them is 2, 
' what is the other number? 

3. If x stands for the total number of men in a regiment, 
‘and one-tenth of the regiment increased by 5 are sick, what 
| expression denotes the number of sick? 





Mg 
%. 


48 EQUALITIES [Cuap. IV. 


4. Write the algebraic expressions for x increased by m, 
decreased by m, multiplied by m, and divided by m. 
5. The numbers 1, 2, 3, 4, . . . are consecutive integers, 
How much greater is each than the preceding one? 
6. If n represents any integer, what will represent the 
next consecutive integer? 
7. If n represents an integer, what will represent the next 
preceding integer? 
8. If n is the middle one of five consecutive integers, what 
will represent the other four? | 
9. The numbers 2, 4, 6, 8, ... are consecutive even 
integers. How much greater is each than the preceding? 
10. If n represents an even integer, what will represent the 
next consecutive even integer? 
11. If n is the middle one of five consecutive even integers, 
what will represent the other four? 
12. If two numbers differ by 10, and the greater is x, what 
is the other? 
13. If A has $100 more than B, represent the money of each 
in terms of x. Do this in two ways. 


Hint: First, let c = B’s money. Second, let x = A’s money. 


14. If A is 5 years older than B, and B is x years old, 
represent their ages in terms of x (a) at present; (b) in 10 years 
from the present date. 

15. Two men divide $1000 so that one shall have four 
times as much as the other. If ¢ is the smaller share, represent 
the larger share in two ways. 


PROBLEMS 


1. A rectangle is twice as long as it is wide. The perim- 
eter is 180 feet. What is its width? 


Hint: Let the rectangle be x feet wide. 


2. The sum of two numbers is 18, their difference is 4., 
What are the numbers? 


eer. 41) PROBLEMS 49 


/ 3. Two men are to divide $1000 so that one shall obtain 


‘4 times as much as the other. What should each receive? 


4. Divide the number 90 into two parts which are to each 


‘other as 2 is to 3. 


V5. The sum of two numbers is 9, their difference is 15. 


What are the numbers? 


6. The sum of two numbers is a, their difference is 6. 


What are the numbers? 


/7. The sum of two numbers is 32 and their difference 


Is —4,. What are the numbers? 


8. If n represents an integer, what will conveniently 


‘represent the sum of this integer and the next consecutive 
‘integer? 


9, Find three consecutive integers whose sum is 48. 
10. Find two consecutive integers whose sum is 193. 
11. A rectangle is 10 yards longer than it is wide. “Its perim- 


-eter is 80 yards. Find the dimensions. 


12. A rectangle is three times as long as it is wide, and the 


perimeter is 128 feet. Find the length and width. 


\J8. The United States has 56,000. more miles of railway 


than Europe. The two together have 409,000 miles. Find 


the mileage of each. 
\%4. Three boys together have 120 marbles. If the second 


has twice as many as the first, and the third five times as many 
_as the first, how many has each? 


~j5. A farmer has three times as many hogs as horses, and 


twice as many sheep as horses and hogs together. If there 


‘are 120 animals in all, how many are there of each kind? 


vis. A plumber and two helpers earn together $7.50 per day. 


How much does each earn if the plumber earns four times as 


- much as each helper? 


7, The sum of three consecutive integers is 78. What 


~are.these three numbers? 


g. A merchant owes A three times as much as he owes B, 
he owes C twice as much as he owes A, and he owes D as much 


‘ 


‘ 


50 | EQUALITIES [Cuar. IV. | 


as he owes A and B together. If the sum of his indebtedness 
to A, B, C, and D is $14,000, how much does he owe each? 


19. The length of a field is 3 times its width, and the dis- 
tance around the field is 200 rods. If the field is rectangular, _ 


what are the dimensions? 
\20. If three times a certain number is added to twice the 
number, the sum is 35. What is the number? 

21. A real estate agent purchased 3 houses, paying twice 
as ouch for the second as for the first, and four times as much 
for the third as for the first ; if the difference of the cost of the 
second and third is $3000, what is the cost of each? 

22. A room is 15 feet long, 14 feet wide and the walls contain 
464 square feet. Find the height of the room. 

23. How much must be invested at 6 per cent simple interest 
to amount to $650 in 5 years? 

24. Two motor cars can run one at 20 miles an hour and the 
other at 25 miles an hour. If the faster car sets out to catch 
the slower when the latter has 15 miles start, in how many 
hours will it catch up? 

25. A golfer knows that it is 380 yards from the tee from 
which he starts to the hole. After playing two strokes with 
the brassie and one with the mashie, he finds the ball 5 yards 
short of the hole. Assuming that he played in a straight line, 
and that each brassie stroke is twice as long as a mashie stroke, 
what is the length of each stroke? 


‘Arr. 41] REVIEW EXERCISES AND PROBLEMS 51 


REVIEW EXERCISES AND PROBLEMS 


1. Give a factor of each of the following and state its coefficient : 
3mn ; ax; 49xyz; bez. 
| 2. Write the following expressions using exponents: ©@-@-@-Yy- 2-2; 
a 3-3-s-s; 125;10000; m-n-x-m-n-n-e. 

3. Write the following expressions without using exponents: 2a’; 
Gy ab’; 3 xy 

4, Add on the number scale: 5+ (-—2)+3+4+(-9) +24 (-2); 
—-§+64+24+(-5)+7+(-6). 

5. Subtract on the number scale: -—4 — (-4) -6-1-(-9) 
We( —2); 8 —(-8) —7 —(-2) -10-(-1). 
6. Show on the number scale that —2 — (+4) = —-2+4+(-4); and 
B- ( —4) =8 + (+4). 
| 7. Write a formula giving the cost c in dollars, of 40 chickens, aver- 
‘aging p pounds, at r cents a pound. 
i 8. Write a formula which gives the cost c in dollars of n miles of wire 
‘fencing at k cents a rod. 

9. Write a formula which gives the cost c in dollars, of m miles of 
wire at d cents a pound, if one rod of wire weighs p pounds. . 


Perform the following multiplications and divisions : 





—18 100 
4 10. -2-3- —5. 14, <——, 
| 2 ees Tecan 
ae certls ° 5 0 ‘ 6 4 3 
11. -1- -17- -2. 156.. =—_—- : 
( 5 19. =5 
| peer ee it. 1. go a tlc 99, 2+4. 
| —2--2:-3 4 
13. 10-0-—10- 2. 17. 22:0°3 91, 228 +8: 12, 
2--3 ce ak ah 


22. Define and illustrate the terms power, exponent, coefficient, abso- 
lute value. 

23. If a is greater than b, is the absolute value of a greater than the 
absolute value of b? Give examples. 

24. Give the results in the following: a+0;a-0; a-:1; a+1; 
a-0; O+a. 

25. How many values of x satisfy 2x = 3x —2x? 

26. How many values of x satisfy 2x = 2x — x? 

27. Distinguish between an equation and an identity. 





52 EQUALITIES LCuap. IV. 


28. The equation x — 5 = 1 is satisfied by x =6. If 4 is added to 
the left member and 5 added to the right member, is the resulting equa- 
tion satisfied by x = 6? \ 

29. State four principles used in solving equations. 

30. Supply the missing term which makes the equality 


5a +6-—2 =144+?-1244422 
an identity. 
31. Find the corresponding values of .5% + 2 when z has the values 
0, 1, 2, 3, 4, 5. Represent these values graphically as in Art. 18. 





See 
32. Find the corresponding values of : when z has the values 


+2 
La hit hy Saeee 
33. Which of the principles referred to in Exercise 29 are used in 
obtaining the second and third of the following equations : 


If 2 of A’s money = $400, (1) 
then 4% of A’s money = $200, (2) 
and of A’s money = $600. (3) 


384. Upon what principles does transposition depend? Illustrate. 

35. The beam AB supported at its 
: center (see figure), just balances when 
one end is weighted with x + 5 pounds 
and the other with 12 pounds. If 5 
pounds be taken from the left side, how 
much must be taken from the right side 
in order that the balance be maintained? Hence, x pounds balances how 
many pounds? 

36. Suppose the left pan of a balance contains x — 7 pounds and the 
right 24 pounds. Let 7 pounds more be put into the left pan. Then what 
must be done to maintain a balance? Hence, x equals how many pounds? 

37. Illustrate with the balances the processes performed in solving 
(a) x+9=12; (0) 2x-4=2+46; (c) 12-2 =17 — 22. 





CHAPTER V 


ADDITION 


_ 42. Terms of an expression. An expression may contain 
one or more + or — signs which separate it into parts. Such 
a part with the sign preceding it is called a term. Thus, in the 
expression, ab + 2xy — 7c, the terms are ab, 2xy and —7c. 


43. Monomials and polynomials. An algebraic expression 
‘of one term only is called a monomial. Thus, 27, 5ab and abcd 
are monomials. 
_ If an expression contains more than one term, it is called 
‘a polynomial. For example, ab + 2ry —7c, 3x — 3y, and 
a+b+care polynomials. In particular, a polynomial of two 
terms is called a binomial, and one of three terms, a trinomial. 


44. Similar terms. Terms that have a common factor are 
called like or similar terms with respect to that factor. 

Thus, 2a and 7a are similar with respect to a; —52*y and 
8x*y are similar with respect to xy ; 6abx and 5mnz are similar 
with respect to 2. 


EXERCISES 


With respect to what factors are the following pairs of terms 
similar? 


aeenr 12x. abe, xyz. 
hema, Cams". abc, ax*y. 
—ta’mn, 5mnv'. 


36a2m5t*, —at*m?. 


2 

3. —4abx, Zamy. 
ew ae a 
cas 4 - or, 13 - 8. 


oO OM AD 


53 


54 ADDITION [CuHap. V. 


45. Addition of monomials. We have already seen that 


2a + 3a = 5a. 
Similarly, 
7x + 4a + 6x = 172, 
and 
ax + bau +.cx = (a+b+4+c)z. 


The method is the same if some of the terms are negative. 


Thus, 
8a +a — 5a — 2a = 2a, 
and 
am — bm +em —km = (a—b+e-—k)m. 


These examples illustrate the principle that the sum of like 
terms is the product of their common factor by the sum of its 
coefficients. 


EXERCISES 
Name the common factor and find the sums of the following : 
1. 12a and 4a. 13. 2b, 9b, and —7b. 
2. dy and y. 14. 8d, 0.7d, and d. 
3. 8x and 2172. 15. 1.6m, 3m, and —3.5m. 
4, —92x and 142. 16. az, —bx, and —cz. 
5. —5r and 5r. 17. am, —2m, and m. 
6. 207? and —7r’. 18. abc, xryc, and —2zyc. 
7. 0.5m and 2.3m. 19. 22°, 1023, and —12z’. 
8. 3-7 and 9-7. 20. 1r?, r2, and 27": 
9. 4-25 and—4- 25. 21. 27r, 9rr, and —127r. 
10. 6. —3 and -10--3. 22. >, i and - 
11. —2n? and —7ni. 23. :, a and al 
12. 8- 3? and —3- 37. 24. sa ze and =) 
eee: n 


25. 3(a+y), —7(~@+y), and 13(x +y). 


Arr. 45] EXERCISES 55 


26. 30(5a — 4), —9(5a — 4), —27(5a — 4), and 11(5a — 4). 

27. a(x +y) and bia + y). 

28. a(c — 4), 7(c — 4) and —3(c — 4). 

— 29. Let xy be represented by one unit on the scale. Add 
on the scale day, —10xy, 8ry, and —2xy. 

30. Find the sums in the first five exercises by adding on 
the scale. 

31. How much richer am I at the end of the day than at 

the beginning, if (a) I earn 3a dollars and spend 2a dollars? 
(6) I earn 2a dollars and spend 3a dollars? 
32. One city lot contains 16x? square feet, a second lot is 
82 feet long and 9x feet wide, and a third lot is 100 feet long and 
AO feet wide. How many square feet in the three lots? How 
many square feet if « = 12? | 

$3. If ¢ = 10, what are the values of #, #,-é*, é, ¢§, 
and ¢7? 

_ 84. Find the value of #+8+é+t+1,if¢ = 10. Also the 
value of 3¢4 + 6@ + 5 + 2+ 7, and of 9¢° + 90? + 9. 

35. We may write 427 = 400 + 20+ 7 = 44 2¢+7, if 
t=10. Write in terms of ¢ the numbers 8699, 2941, 501008, 
and 345678. 


242-431-110 -9 -8 7-6 -5 -4 -3 -2-1 012 3 4 5 6 7 8 9 10 11 12 
Fic. 9 
36. Show on the number scale that 5+3=3+45. Show 
that a + b = 6 +a when a and b have the sets of values 4, 7 ; 
~3,8; —5, —6; 10, -13; —12, 12. 
37. Show on the number scale that 


(8 4+5)+2=3+(5 +2). 


38. Show that (a+b) + c=a + (b+ c) when a, 6 and c 
have the sets of values 1, 4,7; 6, -38,8; -—3, -—7, -2; —5, 
8, —3. 


, 


56 


ADDITION [Cuap. V. 


46. Simplifying polynomials. The terms of a polynomial 


may be arranged and grouped in any manner without changing 


the value of the polynomial.* 

By the use of this principle we are able to reduce many 
polynomials to simpler form. The result in the following ex- 
ample is found by first rearranging the terms and then adding 
together like terms. 


8x + 2Qy — 3x + Ge — Ty + 82 = 8a — 3a + 2y — Ty + 2 + 32 


= 5a — 5y + Ye. 

EXERCISES | 
Simplify : 
1. + 4y — 2x + 5a + Oy — y. 
9°. 364 BF 6G ads 
3. 2.503 — 7a + 8a? — 3.50? — a? + 4a. 
4.A—-B4C0 —2A43B —C: 
5. 3m? — bn? + Tm2n? + 8n? — m’. 
6. 4-5-8:-843- 5 — 0. 
73-8446 — O- one: 
8. 1q 4+ 3m — 4k + 3k — 4a + gm. 
9. 247? + 35r + .6r? — Ir. 
10. m— Ta n+5 
11. Gry + 3ay? — vy — 40y? + Cte 


| Sarr? + Sarr? — Sar? + Tr’. 
-3.11-5-546-5-9:-11l —=5 4 


* This principle includes two fundamental laws of addition : 

(1) The commutative law, or law of order, which says that 
a+b=b+4a. 

(2) The associative law, or law of grouping, which says that 
at+tb+c=a+(b+0c). 


Thus, 8+5-3+11=5+(11+8-38) =(-3+8) + (11 +5) =21. 


Similarly, 2a + 6a — 15a = 2a + (6a — 15a) = —15a + (2a + 6a) = —Ta. | 





Anrs. 47, 48] ARRANGEMENT OF TERMS 57 


14. 4a — 4b + 4c — fa + 4b — $e. 

15. 5ab + 8ac — 7bc — 2ab + 9ac + 2bce — 2ab. 
16. ab" — 3a2"b" + ab” — 5a"b™ + Tab" — ab". 
17. m* — 2n? + 5n® + 38m? — 4m? + n? — nn’. 

18. 2(a+b) +5(a +). 

19. 3(~ — y) + 9(¢ — d) — (wx — y) — 4(c - d). 
20. (m+n)? —7(m +n) + (m+n) + 6(m 4+ n)?. 


4%. Arrangement of terms in a polynomial. The polynomial 
2 + 3x72 +. 3” + 1 is said to be arranged according to the de- 
scending powers of x; 7 — 38y + 7y? + 6y? is arranged accord- 
ing to the ascending Sirians of y; and 523 4+ 4a’y — 82y? + y' 
is arranged according to the descending powers of # and the 
ascending powers of y. 

In the addition, subtraction, multiplication, and division of 
polynomials which contain different powers of the same letter, 
it is usually advisable to arrange them according to the ascend- 
ing or descending powers of one letter. 


48. Addition of polynomials. In the addition of polynomi- 
als it is convenient to write the terms in columns, each column 
containing only like terms. 


Example. Add 2x +y + 5z, 7x —y — 2z, and —4x + 3y —7z. 


SOLUTION: Cueck: Let + = 1, y=1,2=1. 
2e+ y+ 52 8 
7x — y—2z 4 
3 —4x + 3y — 72 -8 
5a + 38y — 42 + 


A check on an operation is another operation that tests the 
correctness of the first. 

The above solution may be checked by substituting any set 
of numbers for the letters. Let x=1, y=1,andz=1. The 
values of the given polynomials are then 8, 4, and —8, and the 
‘sum of these values is 4. Since the sum thus found is the same 
as the numerical value of the answer for x = 1, y = 1, 2 =1, 


| 


3% 


58 ADDITION [Cuar. Ve! 


the answer is said to check. Such a check, though not strictly 
a proof that no mistake has been made, shows that the result 
is probably correct. Failure to check shows that a mistake 
has been made. 


EXERCISES 
Add: 
1. 304+ Ty + 92 6. 2003 — 4a’%y + 12ay?+ y° 
8a +4y+ 2 —323 + Bary — 84ay? — 5y? 
Or — 5y — 82 go-— sy+t+ aye 
2. 8a — b+ 2c 7 1.6m?— 3mn + 3.05n? 
7a+4b- ¢ —.9m? + 4mn — .45n? 
a- b- ¢ L.im+ mn — 2.25n? 
3. 1872 -— 7x+3 8. 4(a —b)? — 8(a —b) +15 
—477+ 2-9 21(a — b)? — 11(a — b) — 27 
102? — 14% 47 —14(a—b)?+ (a-b)+ 4 


?(a—b)?— ?(a—b)— F 


4. 44— 3B +200 9. 5(2 —6) + 12G7 79) ae 
“9A 13 B19 (x — 6) — 30(y +9) -— 2 
5A a1 Bee — 2(x — 6) + 5(y +9) — 82 


6. st — | ee 
wterty—zZ 
—w—-x—y-z2 
10. 6(m — n)? — 9(m — n) + 11, 8(m — n) — 27(m — n)?, 
43 + (m — n)? — (m — n), 9 — 9(m — n)?. . 
11, 2¢— 30° +22 — 5e+8, 62! — 6a? +11, a + 142-7, 
lL —a2+ar+ 2 -— %, © + J, 
12. 3ab — 4bc + cd, -—ab+6cd +12be, ab — be, 
ab + be + cd. 
13. 62? — 42%y + bay? — y', —2a2 + y®, Gay — 12ay? + 3y’, 
1423 — la’y + ry? — dy’. 
14. ir —45 +t, 2t —4s — 1, 23s — it, 8r, —és, t. 


‘Arr. 48] EXERCISES 59 


m0. 270° — 40? + a— 10, 2a — 12+ 9a? — a’, 18 + 5a’, 
6 — 3a° + 2a? — 9a, a — 2. 
We 16. 5(k + 1)? — 2(k + 2) — 7, 20144 +12? —-(k+) +5, 
—6(k +124 3(k 41) - 9. 
17. 3" — 371 + 6, 5-3" + 4.387"! — 1, —2.3" — 4.371 + 8, 
18. 4a —b + to, 2a + $b — $c, a—b—c, —3c + $b — 4a. 
pear 32 41, tt et + 
We —¢+32-8%, 41,17 -t-P-#-t-#, 
, 20. a + .1b + Olc + 001d, d°— 1c + .01b — .001a, 
la—b+c-—.ld. 
_ 21. a+b, c—d, d—a, a+b-—céd, b+d, -a-e, 
im d +5, a, }, ab, c. 
ime 22. (a+b) + (c — d) — (¢ +f), Bia b) —tliced).. 
Bets), (¢-d —2a+b) - Ue +f). 
23. .lm+ .2n+ .3p + .4q, Olg — .02m + .03p — .04n, 
(001n — .002p — .003q — .004m, .085m — .167p — .225¢. 
94. Find the sum of P and 3 times Q, when 


P=2 + dry +327 + 9’, 
and 
Q = 22° + Sa’y — 4ry? — y’. 
25. Use the same values of P and Q as in the last exercise, 
and find the value of 2P — Oavhen 2 = 1, and y= 92: 
96. Write 236, 327, and 413 in terms of ¢ = 10, and add. 
(See Exercise 35, Art. 45.) 
| 27. Write 8964, 50231, 100000,. 847, and 2140 in terms of 
‘t= 10, and add. 
mas. Add: a®* + 3a%b" +'3a"b + b*", 8a*" — 8b", 
mr” — 1207"b" + 6a" b — b%, — a + 9a2"b” — 27a"b?” + 27b%". 
29. How much is the sum 2 + y changed (a) by increasing x 
by 1, 2, 3, 4, and so on to 10? (6) By also increasing y in the 
same Taney: (c) By increasing x and decreasing y? 
' 30. Answer the same questions as in the last exercise for the 
. difference 2x — y. 7 





CHAPTER VI 
SUBTRACTION 


49. Subtraction of monomials. As shown in Art. 28, a 
number a may be subtracted from a number b by changing the 
sion of a and adding the result to b. 




















Example 1. From 10z take —6z. 
SoLurion: 10x —( —6x) = 10x + ( +6x) = 16x. 
Example 2. From —8a take —4a. 
So,uTION: —S8a — (—4a) = —8a + 4a = —4a. 
Example 3. Perform the subtraction —4zxy 
SoLution: —4zy — (+32ry) = —4cy — $ry = —'y' xy. 
EXERCISES 
Subtract : 
1. 132 7. 14s 13. 7(x — y) 
4x —8s —(% — y) 
2. 20ab 8. abe 14: Tignes 
—5ab —4Aabc PET ahs 
8. —d5xy 9. —-2x 15. 11:10 
5a*y 9x —6° 10 
4, —122° 10. —wv 16. 7(x +a) 
—42%3 16u7v 26(a + a) 
5. —8mn 117 ame 17. —13(p? — q?’) 
—4Amn ym 7 —20(p* — ¢) 
6. (=) 20D" 1 PZ 18; 22540 


P| 200" —0.dxryz 3° 5 








\ 
f 


Ars. 49, 50] SUBTRACTION OF POLYNOMIALS 61 























19. 12°5m 26. «+ 4y S3eeeo 
—12:5m —sy —b 
20. a+b Ze 1k 34, 57" 
EB & oe 
21. 3x -—y 28. 7x 35. 8ab 
= 7T y Sa 
22. 5a —™m 29. Q9v SGU 
—m 9 ag 
23. a—2 30, 12 37. m—n 
: ix kk m 
84. br +5 31. 28 86. ees 
10 oe s 
25: 3-2 SPP lie bf es Bars 
—5 oe pases 








90. Subtraction of polynomials. In the subtraction of 
yolynomials like terms may well be written in the same vertical 
‘column. If several powers of one or more letters occur, it is 
ionvenient to arrange the minuend and subtrahend according 
o the powers of a particular letter. 


Example. Subtract 3x3 — 7 — 52? — x from 1223 + 82? -— 4x 4+ 2. 


SOLUTION: CuHeck. Let z= 1. 
127? + 82? —47+2 18 
oa? — $2? — 2-7 —10 
Oz? + 132? — 324+ 9 28 
Beitract: EXERCISES 
1, 1027 — 37 + 12 3. 2 — 9y— 2 
827+ 24 —-— 9 x — Illy + 52 
| 9, 5a + 4y — 3u 4, 3c — 42% — 8y3 


2x — 3y + 9u 32° + 4a°y — 8y 


> |. 
} 


62 SUBTRACTION (Cuar. VI. t 
Bi. gb eee ae 7. @-sbee 
—a + 8b — 5c — 8d —~at+bv+2 


6. 12ab — 3cd4+ def 8. 6 (x + y) — 2(m? + n’) +a 
—ab + 17cd — 50ef Q(x + y) — 4(m? + n*) — U 

2a +y) + 2(m? +n) +o) 

9, 27(a +b)? + 5(a +b) - 9 
(a+b)? —5(a+6) +1 








10. (m +n)? + 3(m +n)? + 3(m +n) +1 
— (m +n) + 2(m +n)? — 2(m +n) — 1 


11. From 623 — y3 — 3a’y — 4ay” take bay? — 3y° + 2x3 — 92*y. 
12. From 7a™ — b” + 5c? + 11d? take d? — 5cp + 5a™ — 2b". 
13. From 8(a + b) + 12(@@ + y) -2 take 5(a + b) - 
A(x + y) + 102. 
14. From 6(r +s)? — 7(r +8) +25 take 5(r + s)? — (r +8). 
15. From 9(x —1) + 5(y4+ 2) - 4(z — 3) take 
O(a — 1) — 3(y + 2) — 9 - 3). 
16. From 10(a — b) + 16(¢c — d) +e take 6e — 4(a — b). 
17. Subtract 723 — 3y? — 2a°y — 7xy? from 7y? — 2 + zy” 
— xy. 
48. Subtract 3ab + 2a%b? from — 7ab. 
19. Subtract 2(2 + y) + 5(@ + y)? + 14 from 10 + “st. 
20. Write 8639 and 27941 in terms of ¢ = 10, and subtract 
the first from the second. : 
21. From 5:-72—4:-7+410-7 —7 take 4-73 + 7 — 6: 7. 
Simplify the result. 
99. From 2-21 +3-31+4-41 take 5-41 —6-21+431 
Simplify the result. . 
23. How much greater is 7° + 2° + x than v3 + 2? + 2? 
24. How much greater is 2* than 2? Answer this ques 
tion when x has the values 4, —3, 0, and 4. What is the mean. 
ing of a negative answer here? 


er. 50] EXERCISES 63 


25. Find a value of x that will make 62° equal to, one that 
“ll make it less than, and one that will make it greater than 
2. 

26. How much greater is 0 than —11? Than 3x2 — 40? 

27. From the sum of 5a? — 22ab — 38b? and 42a? + 19ab take 
5a + 27ab — 50b?. 

28. From 5m’ — .05m? + .7m — .001 take 
tom? — Am? — 3.2m — .31. 

29. What must be subtracted from 4b — 4b + 2c so that the 
smainder is 2a + 2b — 4c? 

30. What subtracted from 22 will leave — x4 + 4a3 — 12x? — x? 

31. What subtracted from x + y +z will leave 10x + 100y 
- 10002? 
~ 32. From 2" + 6x" y —182"y? + 7y® take y® + 4a"—ly — 52x 
i: 4". 

33. From 3-2” —6"+7-4"4+ 9.5" take 2” — 5.4” — 57, 
_ 34. Take x* — 3y? + 7xy — 4 from the sum of 32? — y? + 13 
Od ry — x? — 4? — 4. 
85. Take 2k? + 5h? + 11hk —-3h+4k from the sum of 
+ 4k —h? —k? +1and k? + 5h? — hk + 38. 

36. Take the sum of m? — 3m?n + 3mn? — n3 and 


2 
en + > — n>+ 3 from 3m? — 3n’. 


37. Take the sum of — 2p‘ + 5g4+ r4 — per +1, 
ogr — 3p* + q—r', and p*+q*+ pqr—4 from the sum of 
0* + 3q* — r4 and g‘ + par + pt + 3. 


CHAPTER VII 
PARENTHESES 


51. Removal of parentheses. The expressions 
6 + (7 +4) and 6 +7 + 4 are equal. 
Similarly, 9 + (6\— 2) = 9 eee 
In the same way, a+ (b-—c) =a+b-c. 


These examples illustrate the principle that the value of an 
expression is not changed by removing parentheses preceded by « 


plus sign. 
In subtracting the sum of 3 and 5 from 12 we may write 


12 — (3 + 5) or 12 — Sia 


since we get the same remainder, 4,in each case. In subtract 
ing the difference, 7 — 2, from 19 we have 


19 (7 = 2) =19 — 


since the value of each expression is 14. 
In general, such an expression as @ — (b — c), means tha 
b — cis to be subtracted from a. Hence, Art. 27, we chang 
the signs of the terms of b — ¢ and add. 
This gives, 
a—(b—c) =a+(-b+e)=a—-b+e. 


These examples illustrate the principle that parentheses pr 
ceded by a minus sign may be removed by changing the sign of eac 
term within the parentheses. 

Expressions often occur with more than one pair of pare: 
theses. When one pair occurs within another pair, bracke 


and braces may be used to avoid confusion (Art. 15). Allp 
64 





Arr. 51] REMOVAL OF PARENTHESES 65 


ventheses may be removed by first removing the innermost pair 
vecording to the principles stated above; next the innermost 
jair of all that remains; and so on. 

Example. Remove the parentheses in 


2a — {6b — 2c + [a — (b -c)] + 2b}. 
SOLUTION: 


ta — {6b - 2c + [a — (b-c)] + 2b} = 2a - {6b — 2c + [a—b +c] + 2} 
= 2a — {6b -2c+a-—b+c+42b} 
= 2a — 6b + 2c -a+b—c—2b 
= a—Tb+e. 

CHECK: Let a =1,b =2,c=3. Then, 

2— {12-6+ [1 - (2-8)] +4} =1-1443. 
-10 = ~—10. 
EXERCISES 

Remove the parentheses and simplify the results by collect- 

ng like terms: 

3a + (2a — 4). 

5x + (7 — 2). 

2x +1 —- (4 +7). 

b — (2a — 3b) +a. 

n—-1-—(1—7n). 

n—-1—(n+1). 

a+b—c—(c—b-a). 

2x + (3x — 5z) + (52 — 2x + By). 

1- {1+ [f1-(1+1) -1) 41} -1. 

(3a — 4b) — (a-—6 +4 2). 

. £— (2+2-—- (82 — 7)]. 

7 . m— {3n + [2m —n — (5m —n+6)]}. 

. . 63 — [80 — (15 — 4) + 2] - 1. 

. 4x — [5a + (v — y) — y] + 4y. 


a ee ee eee 


ee e ee 
PrP oD BF © 


66 


15. 
16. 
17. 
18. 
19. 

20. 


PARENTHESES [Cuar. VIL , 


12ey + 3y? + [6ax — (2b + 7az)]. 

(22 —y +7) - [x- (y - 2)]. 

4a + 3b — [ex ta+b—2y-(e+y)]. 

atm — ([6 + 4a — (6m + 2)] — (7m — 4a = 8)}. 
a—b— {(a—b) - [a—b — (@—b) - b-a)] —b +4}. 
p+q—r—([(p-—at+r) -(-pt+a-*l. 


Solve the following equations : 


91 Or mela) pee 

22. a + (2a — 1) = 59. 

93. (32-—4) —2 =2+4 6. 

94. + — 24+ (382 — 9) — 15 = 0. 

25. (6m — 3) =9 — (4m + 12). 

26. 22 —14 — (a — 12) = 6. 

97. 4t--2=5-— (8t+7). 

28. .5a2 — (.0la + 9) = 40. 

29. Show that x — (m+n) =x —m-—n when z, m, and n 


have the values: 5, 1,2; 2,9,4; 0,3,3; —2, 2, 9. 


30. Using the same sets of values show that 
a—-(m—n) =x2—-mM+N. 
31. Using again the same sets of values show that 


x — (m +n) is not equal to x — m + n. 


For what particular value of 7 is 


a—(m+n)=x-—m+n? 


52. Insertion of parentheses. Terms may be inclosed in 


parentheses with or without changing their signs according as 
the sign before the parenthesis is minus or plus. . 


Examples. e-2+b-y=xr2-(2-b+y). 
e+3—-b+a=2+(3-b+¢C). 


That these results are correct may be shown by removing the pa: 


rentheses, 


Arr. 52] EXERCISES 67 


EXERCISES 


_ In each of the following expressions inclose the last three 
erms in parentheses: 


lL at+b—c+d. 7 —xX—y—Z2-W. 

» ab+pq —rs — xy. 8. a+b-—c-—d. 

3 4—-2a+b0-y. 9. 1 — 4m? + 4mn — n?. 

ma? — & + 2bc — c’. 10. a+b+c-—ux-y-z. 
he ee = ty — 4. 11. 36 — 9m* — 6m?n — 1. 


—2Qey+y?—a@-—2ab-—b% 12. 4a%4+274+1. 


we 
. 


| 13. A rectangle is x + 20 units long and x + 7 units wide. 
Nhat is its perimeter? 

14. Three points A, B, and C are in a straight line, and B 
s between A and C. If the distance from A to Bism +n+6 
nehes, and from B to C is x — y +7 inches, how far is it 
‘rom A to C? How much farther is it from A to B than from 
Beto C? 

15. Write the remainder when the square of x plus the 
wroduct of « and y is subtracted from the cube of z 
ninus 7. 

16. By what amount is m? greater than 2m — 1? 

17. By what amount is a — 3 greater than 2a — b? 

18. What is the smaller part of x* if x? + x — 1 is the larger 
dart? 
19. By what amount does one million exceed 
+ 544+ 8842+ 71+ 5 if ¢ = 10? 

20. What number is 4¢ +8 greater than — 47? Find the 
‘umber when t = 10. 
| 91, What number is 3d? +¢-+ 1 less than 300? Find the 
tumber when ¢ = 10. 

22. Write in the form of a fraction the quotient of 
Ww + 12t + 4 divided by 3¢ + 2. Find this quotient when 
= 10. 





68 PARENTHESES _  ([CHar. VIB 


23. What is the error in the statement 7-y+2—-W= 
g—-(y+z—w)? | 
24. What is the error in 


ax + by — ay — ba = (ax — bx) — (ay + by)? 


53. Collecting literal coefficients. 


Add : 
ih ax 7. —6z 12. av? — 3mv 
bx — max —3v° + 
(a + b)x 
2. cy 8. 3x? + bx 13. gt — ar 
— dy ka? + 9x —3t — br 
(c — d)y 
3. —Ax 9. ap? =p 14. 42? — 7z 
vr —4p? + Pp ba? — az 
—(a + b)x 
4. am 10. ax — by + cz 15. 3c? + 2d 
bm —2x%-—dy+ 2 Ic? + nd 
5. 3m 
am 11. be—ay+dy 16. 390? — igt 
: —2be + xy — dy 29? + gt 
av <3; | ne Ti ae 


CHAPTER VIII 
MULTIPLICATION 


54. Products of powers. It follows from the definition of 
an exponent, Art. 11, that 


a=a-a, 

G=4-a*a*a, 
and hence that a?-at =a-a-a-a-a:a=a' = @*4, 
‘Similarly, Cee ea ee ee EL eH ee ele: 


These examples illustrate the following rule for combining 
exponents in multiplication : 
The exponent of a letter in a product equals the sum of the 
exponents of that letter in the factors. ; 
_ This rule may be stated in algebraic symbols in the form, 


anm-an=anrtn, (1) 


EXERCISES 


1. Use the definition of an exponent and show the sree 
Mes c*; a7b?; Garty; 51?m?n; a”. 
Complete re following tidieated multiplications : 


2. a?-at-a®, 8. (x)? 14. 102-108, 20. a?- a”. 

am b*-b- bd. 9. (m?)3. 16504). Ort be 
4m-m-m'. 10. 3-3-3°. 16. (3). 22 
mye. 7?-y. 11. (2°). 17. -($)?. Osa i 
Wee 2°, 12. —24.25.28 18. (3)-(3)% 24. (-a). 
| o —r°- 2°, 13. r8.72-79, 919. 20-208. 925. —a?- (—a)?. 


26. Verify that a-b = b-a, when a and b have the values : 
N@7, 54; 18, 1.8; 229, 11; 341, 2.56. 
69 





70 MULTIPLICATION [Cuap. VIL | 


27. Verify that a-b-c =a-c-b =c-a-b, when a, b, and 
c have the values: 3, 4,5; 25, 35, 41; .1, —%, .2. ! 

28. Verify that (a-b)-c =a-(b: BY ener a, b, and c have - 
the values given in the last exercise. | 

29. What must be the value of m so that (a) m-5-6 = 15; : 
(b) (m-6)- (3-5) = 180; (¢c) .2-6-m- 4 = 96? 

30. Show that 5a*y’- 204 y = (5-2): (a3- a*)- (yy), when 
a =1 and y = 2. 

31. Show that —722yz - 3ay%z- —x4y® = 212” y®2?, when x = 1, 
y = 2, and z = 3. | 

32. Show that (a”)" = a™, when a = 5, m = 3, and n = 2. 

33. Show that (a™)"=a™ when m and n are any) 
positive integers. | 


Hint: (am)n = am-qm-aqm.,.. ton factors. 


55. Products of monomials. The factors of a monomial 
may be arranged and grouped in any manner without changing 
the value 6f the product.* Thus the order of the factors may 
be changed, asin 5-6 =6-5. Also, the factors may be grouped 
in different ways, as in (8-5)-8 =3- (5-8). 

We make use of this principle in finding the product of 
monomials. 


EXERCISES 
1. Find the product of 72? and 82%. 


SoLuTION: 722 + Sat = (7-8) - (a? - x4) (Since the factors may be re- 
arranged and grouped in 
any manner.) 

= 567°. (Using the law for combin- 
ing exponents in multipli- 
cation.) 


* This principle combines two fundamental laws of multiplication : 
(1) The commutative law, or law of order, which says that 
a-b=b-a. 
(2) The associative law, or law of grouping, which says that 
‘ (a-b)-c=a- (0-0). 


| 


Art. 55] 


EXERCISES 


2. Find the product of —3am?n? and 6a3min. 


71 


SOLUTION: —3am?n? - 6a*min® = (-3 - 6) + (a+ a3) + (m2 +» m3) + (n? +n) 


=-—18a‘m?n’. 


Complete the following indicated multiplications : 


1. a’®- ae. 
ate AL: 
- Opg: —g. 
. om-4m. 


. 2a- 5b. 
. 8u2v- —Tur?. 


2 
3 
4 
Dour say. 
6 
7 
8. 12a%b4c® - abe. 


eae yee fiz - 2°22". 


toea—o0d.- —40.- —c. 


14. lla’a’yz- —6y%2?- — 


fee + oe —2- 34r?, 


16. 4°. 7?m?n?.4. 


mag. oF 8°p°- 2+ tp. 


7mn. 


er eo 4 1. 1. 


5 
Bu". 


| 
P| 


19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 


27. 


28. 


29. 


30. 


31. 


32. 


33. 


34. 


35. 


36. 


(x?yz)?. 
(3a°b*c)?. 
boar)": 
(—11m5n8p°)?. 
(—3aa%y®)*. 
(—5p?q’)*. 

4. fa’. 

—5 - 2m'. 


4mn 








—— 


72 MULTIPLICATION [Cuar. VIL. 


56. Multiplication of a product by any number. A product 


is multiplied by any number by multiplying any factor of the 


product by that number. 


Thus, 3.(2-3-5) =6-3-5=2-9-5 =2-3- 15. 
5. (Qry) = 10xy = 2-5x-y = 2a- Sy. 


EXERCISES 


Perform the following multiplications in at least two ways. 


1. 3(4-5). 
6(5- 7). 
10(7 - 8- 2). 
12(4- 4-8). 
A(3ay). 
3(12m?n3). 


HEHE 


Fia. 10 


ST eee 


7. 10(8- 25). 

8. 123(4-5-7). 

9. 334(62y). 

10. —2(—5- 0). 

11. —2(-2-6- —3). 
12. —a(—b- —c-d) 

13. Show that m(x# + y + 2) = 
mx + my + mz, when m, x, y, and 2 
have the values: 2, 3, 4, 5; 3, 9 
—4, 5; 6, 0, —6, —11; 0, 7, 2, I. 

14. Show that m(a — y -2) = 
mx — my — mz, when m, a, y, and @ 
have the values given in the last 
exercise. 


57. Product of a polynomial by a monomial. We know that 
A(5 +7 +3) = 20 + 28 ees | 


This illustrates the rule that to multiply a polynomial by a 
monomial, we multiply each term of the polynomial by the mono- 
mial and add the resulting products. 








f 


i! 
i 
i 


‘Arr. 57] 


| 


diagram. 





POLYNOMIAL BY MONOMIAL 


This principle is stated in algebraic symbols in the form 


a(b +c+d) =ab+ac+ad.* 


The product of two numbers may be represented by a 


Example 1. The product, 3:4 = 
12, is illustrated by Fig. 10. a| ab ac 
d 


h Example 2. The product, , b - Cc 
4(5+7+3) = 20+ 28 + 12, is illus- 


ed by Fig. 11. z eave tt: is 
Example 3. The product, 

a(b+c+d) = ab+ac+ad, is illus- b+e+d 
‘trated by Fig. 12. Fig. 12 


EXERCISES AND PROBLEMS 


Find the following products and simplify if possible : 


i: 


Be 
oo bo 


— 
on 





pt ec) pill ah gl a 


—_ 
ae 


— 
> 


8(5 +4 —7). 
5(7 +9 — 6). 

3(200 + 70 + 9). 
a(b+c+d). 

6(8 + b — m). 

a’x?(a? + 2ax + 2x”). 
5m2n(8m? — 4mn + n’). 
—3ac(a? — 9Yac — c?). 
2y4(a? + 2ry — 6a%). 
—2ab(—a’x + 4ax — 122%). 
(—1): (a — b) + 4(8a — 26 — 4). 


. 8(—-x-—y) —3(24+y — 2). 
. (-3):( —s) + 2t4 8s - 4. 


6(a? + 4a + 4) — 9(2a? — a — 2). 


. 6(7000 + 900 + 10 + 8). 
. 3a"(a2" — 2a"b" + 6"). 
We 


 * This principle is known as the distributive law of multiplication. 


ant2h2n (inh? = 6a2"—-4h3-” ef 1la”—"b). 


74 MULTIPLICATION (Cuap. VIII. 


18. 2°(1 +.2" + 3"). 
19. (x — y)*[(x — y)® — (@ — 9)’. 
20. (a + b)*[(a + b)> — 6(a + B)?]. 

















21. 65 i i). 22. (5 +5- ’). 

SoLurIon: s(F+5-4) - at St SB = 4m + In — p. 

23. 12(75 z ° mn 5 25. 30( FF e s 4 un). 

oy 10 a 9a? fi #), 26. 16(4a + 4b — $c — d). 
Se aadele ham 27. 6(§0 + fy — 32 + 2). 

98. (52-2 BE) 

29 16(7 4 oe . a), 

30. 20( = 4 ue wate 6 1 Sa Ba ) 


31. Find the area of a rectangle whose dimensions in inches 
are 3b and 6b — 7. What is the difference between the num- 
ber of square inches in the area and the number of linear inches 
in the perimeter? Check when 6 = 2. 

32. Find the volume of a rectangular solid whose dimensions 
in inches are h, 2h, and 3h +4. Find the sum of the number 
of square inches in the surface, and the number of linear inches 
in the edges of this solid. Check when h = 10. 

33. Illustrate by a figure that 4(5 — 2) = 20 — 8. 

34. Illustrate by a figure that a(b — c) = ab — ace. 

35. Show that (a + b) (x + y) = ax + ay + bx + by, when 
a, b,x, and y have the values : 1, 2,3, 4 ; 4,7,2,5; —2,3,4, —6. 

36. Show that (m — n) (p — q) = mp — mq — np + nq, when 
m,n, p, and q have the values: 4, 2, 7,6; 1,1, 6,8; 0, —3, 
2, —2. | 


1) ee 
i 

| 
| 


i 
| 


‘Arr. 58] TWO POLYNOMIALS 75 


58. Product of two polynomials. The product, 
(8 + 4) (@ + y), may be found in two ways. 


(8 +4) (@+y) 


Also, 
(8+ 4) (w+ y) 


= 7(" + y) 
= 7x + Ty. 


=3(@+y)+4(r+y) 
= da + dy + 4a 4+ 4y 
= 12 Aye @ 


By using the second method, we find that 


(a+6)@+y) =a@+y)+ba@+y) 
=ax+ay+bxr+by. (See Fig. 13). 


| ‘These examples illustrate 


the following 
Rule. The product of two 


polynomials ws found by , 


multiplying one polynomial 
‘by each term of the other and 
then adding these products. 








| a 


Example 1. Multiply x +3 by x +7. 





ax ay 
Fig. 13 
SOLUTION: CHEcK: Let z =1. 
zx +3 4 
Be ae te. 
x? + 3x 
7x +21 


x’ +10xr + 21 32 


See Fig. 14 for a diagram to 
illustrate this exercise. 


Example 2. Multiply 27? + 3a — 5 by 4a — 7. 


SOLUTION: 
207 + 32 — 5 
Ar — 7 


8x3 + 122? — 20x 


ae oly +55. 


8a3 — 277 —417 + 35 


CHeck: Let x = 2. 


1 


9 


76 


MULTIPLICATION LCuap. VIII. 


EXERCISES 


Perform the following multiplications and check the results 
by substituting numbers for the letters: 


a ee 
COND TS 


i) 
c—) 


oN bs 
wo nw Fe 


bo 
a 


OoONNNDN WN 
ono ont oa 


a ae 
QPP oo Oe ao m wr 


(x+y) (w+ y). 

(x — 4) (24 + 3). 

(a +b) (a — b). 

(80 + 7) (60 + 4). 

(2x + y) (3% — 2y). 

(6m? + 5mn) (m — n). 

(2? +2+1) (e - 1). 

(a2? — xy + y*) (w+ zy + ¥’). 
(a + b)?. 

(a + b)?. 

(30 + 1)?. 

(a+b+c)*. 

(a® + a%b? + b®) (a? — 6). 
(493 — a? + — 1) (2a? — 4x — 7). 


. (800 + 40 + 2) (70 + 6). 

. (xt — Qasy + 2x°y? — Qay + y*) (2? + zy +’). 
(g— 2) (e—3) ae 

. (e +a) (x +b) (x + 0¢). 


(a3 — 30% + 3ab? — 6°) (@ — 2ab + 02); 


. (m — §) (2m — 4). 


1y — ty) (a + 4Y)- 


. (3x — ZY)? 
. (5a — 6b + 3c) (2a + b — 4c). 


(5 6 5) (§+3-5): 
55 9 10) \ ou owe 


. (2m? — mn +n?) (2m — .5n). 

. (1.4402 — .72ab + .09b?) (1.2a — 3b). 
. (3.5m? — 4mn + .15n?) (4m — 6n). 

, (a — Barty + ay?) (w — y). 
(a + ab" + arb + 8") (a — bn), 
(x + yy, 


| 
i 
i 
} 


‘Arr. 58] EXERCISES -- 


31. What is the area of a rectangle that is 32 + 5 units long 
and 4x — 4 units wide? Check when z = 6. 

32. Show that (8 + 3) (7? + 2t-+ 6) = 83: 726 if ¢t = 10. 

33. What is the difference between the squares of two suc- 
cessive integers, the smaller of which is x? Is this difference an 
odd or an even number? 
84. Illustrate the meaning of (a+ 6b+c) (x+y) by con- 
structing a rectangle a+b+c units long, and x+y units 
wide. 
_ 35. Show by a figure how much the area of a square of side 
a is increased by increasing the length of the side one unit. 


' Solve the following equations : 

36. (37 — 2) — 7x — (12 — 3z) = 18x. 

87. 2(4 — y) = 5y — 18. 

38. 3(a + 2) — (a -— 9) = 1. 

39. n+2(n+1)4+3(n+4+ 2) = 91. 

40. 7(4¢ — 3) + (7 — 82) = 1. 

41. How much is the area of a rectangle of base x and alti- 
tude y changed (a) by multiplying its base by 2? (6) By divid- 
ing both the base and the altitude by 2? (c) By multiplying 
the base and dividing the altitude by 2? 

42. How much is the area of a rectangle whose sides are 
v and y changed (a) by increasing both the base and the alti- 
jude by 2? (6) By decreasing both the base and the altitude 
oy 2? 

Draw figures showing the atte products : 





© 4s. (c+y) (e+ y). 47. (a + 2). 
44. (x — 4) (22 + 3). 48. (30 + 1)? 
45. (m+7) (m+ 5). 49. (a+b+4+c)?. 


46. (2x + y) (3x — 2y). 


CHAPTER IX 
EQUATIONS AND PROBLEMS 


59. Equations involving parentheses. 


Example 1. Solve the equation 3(4¢ — 1) — 5(2¢ — 3) = 18. 
SOLUTION: 3(4x — 1) — 5(2x — 3) = 18. 
Multiplying, (12% — 3) — (10x - 15): = 18. 
Removing parentheses, 122-3 — 10¢4+15 =18. 
Transposing and collecting like terms, 22 = 6, 
Dividing by 2, je ao 
CHECK: 3(4°3 —1) —5(2:°3 — 3) = 18, 

33 — 15 = 18, 

18 = 18. - 


In certain equations higher powers of the unknown than the 
first occur, but vanish as in the following : 


Example 2. Solve the equation (x — 5)(# + 3) — (32 —4) = (x — 1)”. 


SOLUTION: (x — 5)(a + 8) — (84 — 4) = @ - 1). 
Multiplying, g? —~27 —15 —384+4=2? -274+1, 
Transposing and collecting like terms, —3r = 12. 
Dividing by —3, z= —4, 
CHECK: (-4 — 5)( -4 +3) — [8: (-4) - |] = (-4 -1)’%, 
9 + 16 = 25, 
25 = 25. 


EXERCISES AND PROBLEMS 


Solve the following equations : 
12D Dee 
4(10 — 2x)°=3(2— 5). 
3(9 — 2x) — 5(2% — 9) = 0. 
7(4y — 3) + 3(7 — 8y) = 1. 
8(3a — 2) — 7x — 5(12 — 3x) = 132. 
6a — 7(11 — a) + 11 = 4a — 3(20 — a). 
78 


EL AH oth eA 


[ 
er. 59] EXERCISES AND PROBLEMS 79 


5(m — 3) + 4(17 — m) = 11 — 7(8m - 6). 
. k — 2(4 — 7k) = 4k — 9(2 — 8k). 
9. 80 — 6(4¢ + 8) = 7x — 3(6r + 1). 
10. 3(y — 10) + 11(2y + 1) = 17(y — 12). 
11. 82(82 + 2) — 27 = 4x(62 — 1) — 147. 
12. (5 — 3a) (38 + 4a) = (1 — 4a) (7 +-3a) - 1. 
13. 21 — 3p(10p + 3) = 45 — 5p(6p — 1). 
14. (2n — 1) (n+ 5) —1 = (n — 6)? 4+ (n 47)? 
15. n?+ (n+ 1)? = (n+ 2)?4+ (n+ 3)2, 


dale, 


— Eee Se Se 


f Peet 5 5-3)- 
16. 4(2x — 3) 6(5 5) = 2. 


piv. 82x — %) = 5 + 5-4) -4(7 - 1). 


4 
18. 4(a — 3)? + 6(a — 2) = (2a — 5)? + 302. 
19. (x-1)?+ 57 + 2) = (© + 1)8 — (a — 4) (22 +7) + 16. 
r 


6 


20. 27(x — 4) — 114( rn 1) S597 ee 6(2p—~ 2). 


21. A line is divided into two parts, one of which is 20 
| Aches longer than the other. Twelve times the shorter piece 
quals 8 times the longer. How long is the line? 

_ 22. A man paid $12.50 for 2 wooden golf clubs and 6 iron 
‘mes. Each wooden club cost 25 cents more than each iron 
lub. Find the cost of each. . 

23. The value of 31 coins consisting of dimes and nickels is 
32.25. How many are there of each? 

24. A tourist climbs from a certain point up the slope to 
he top of Pike’s Peak at the rate of 2 miles per hour, and 
ilescends by the same path at the rate of 4 miles per hour. If 
he round trip takes 6 hours, how long is the path? 

» 25. A man made two investments amounting to $4330. 
Mn the first he lost 5%, and on the second he gained: 12 %. 
Vhat was each investment if the net gain was $251? 






80 EQUATIONS AND PROBLEMS [Cuap. IX. 


In a right-angled triangle the square on the hypotenuse is equal to 
the sum of the squares on the other two sides. The hypotenuse is the 
side opposite the right angle. In Fig. 15, c? = a+b. Ifa=3,andb =4, 
then c? = 3?+ 42 = 25, and c = 5. 
By transposing, we get a? = c? —B? 
and ? = ¢c? - a’. 1 

26. One side of a right 
triangle is 5 inches, and the 
hypotenuse is 2 inches longer 
than the other side. How 
long is the hypotenuse? 





SuaacestTion: Let z = thenum-— 
ber of inches in the unknown side. | 


Then x +2 =the number of 
inches in the hypotenuse, 


and (x% +2)? — a? = 25. : 


27. A rope that is 8 feet 
longer than a flag-pole reaches: 
from the top of the pole to a point on the ground 32 feet from. 
the foot of the pole. Find the length of the flag-pole. | 


Historical note on the equation. The very earliest mathematical 
writer of whom we know, an Egyptian priest named Ahmes, solved equa- 
tions in one unknown. He lived at least as early as 1700 B.C. The un- 
known he called “hau” or heap. One of his problems reads: ‘“‘ Heap, 
its half, its whole, it makes 16.” When zx denotes the unknown, the 





Fig. 15 


equation to be solved is x +5 =16. In Egyptian hieroglyphics this 


tae lame Oo Om Do 


Heap, its half, its whole, it makes 16 
equation is written as shown. The Hindus used the word color to denote 
the unknown, the Europeans early used the word res (thing), and the 
Arabs used the word root in this sense. It seems that next the initial 
syllable of each word was used to denote an unknown. Thus ka (from 
kélaka = black) meant the unknown. It was not until the time of Vieta 
(1540-1603) that a single letter was used for the unknown. The con- 
ventional use of x for the unknown is due to Descartes (see p. 181). 


7 


i 
i 


‘Arts. 59, 60] EXERCISES AND PROBLEMS 81 


_ 28. One side of a rectangular field is 40 rods. The diagonal 
of the field is 20 rods less than half the perimeter. Find the 
area of the field. 


29. The difference between the areas of two squares is 192 
square inches. The side of one is 6 inches longer than the side 
of the other. Find the area of the larger. 


_ 30. One side of a rectangle-is 15 inches longer than the 
yther. The area of the rectangle is 450 square inches . 
greater than the area of a square whose side is equal to 
he shorter side of the rectangle. Find the area of the 
ectangle. 

_ 31. The sum of two numbers is 2. The niffercniom of their 
quares is 80. Find the numbers. 


32. The difference of the squares of two consecutive integers 
s41. Find them. 


33. The difference of the squares of two consecutive even 
‘otegers is 436. Find them. 


_ 34. The radius of a circular flower bed is increased 2 feet, 
hus increasing its area 88 square feet. Find the radius of the 
wriginal bed. 


: : : 22 
The area of a circle is rr?, where r is the radius. Use 7 = 7 


: 


if 


35. The difference of the areas of two circles is 423.5 square 
2et. The difference of their radii is 3.5 feet. What is the ra- 
‘ius of the larger? 

36. Show that (2n)? + (n? — 1)? = (n? + 1)2, and hence that 
n, n* — 1, and n? + 1 may be used as the sides of a right tri- 
ngle. Give n the values 1, 2, 3, . 10, and find the corre- 
oonding values of 2n, aa nh i The formula is known as 
lato’s formula. 


60. Equations involving fractions. We are able to solve 
rtain equations involving fractions by applying the principles 
‘fractions learned in arithmetic. 


82 


EQUATIONS AND PROBLEMS LCuar. IX, 


Example 1. Three-fourths of a number plus 5 equals 47, What is 
the number? 


SoLutTion: Let xz = the number. 
Then, se +5 = 47. 
Transposing, a = 42, 


Multiplying both members of the equation by 4, we have, 


3x = 168. 
Dividing by 3, x = 56. 
CHECK: am +5 = 47. 

47 = 47. 


Example 2. One third of a number plus one fourth of the number 








equals 21. What is the number? 
SoLution: Let x = the number. 
el 
Then, 3 nr = 21. 
Multiplying each member by 12, 4% + 3x = 252, or 7x = 252. 
Dividing by 7, x = 36. : 
CHECK: 12 +9 =21, 
21 = 21. 
EXERCISES AND PROBLEMS 
Solve: 
7. 1 per a! 
1. = =4. pies: aig RE I 
- 6 3 + 6 2 x 
Vi rg Bc 
2. —=3. ~na-5 = WZ. 
y 9 3 
oh iy n—3 
Ss; 9 + 2 =F ise 8. 4 <3 5. 
4 —6 1+ 4 
4. = 2s 9 re ee ae 
8 5 
a 12 
bites Som Lies 10. — =4. 
5 5 +8=13 0 - 4 





| 
: 


i 
i 

i 
i) 
] 


‘Arr. 60] EXERCISES AND PROBLEMS 83 





Bee OF) g 16. ee 
@ 
1 Fac lieee 
— i — —_—_ — —= ese) 
* a3. om aaa 5 = 9) ae 5m. 18. 32 = =. 
2808 ee 
14. 100 + 3000 = 3028. 19. igs 0.004 
| Lote! Susu 
15. ens 20. 5 + 3 = 2s. 


_ 21. An article was sold for $5.40, thus losing one-sixth of 
the cost. Find the cost. 

_ 22. The increase in the value of certain farm land is 14 
times the value 15 years ago. What was the land worth 15 
years ago, if it is now worth $220 an acre? 

| 23. After a decline of 5% in the price of an article, it was 
worth $7.98. What was the value before the decline in price? 

24. The difference between yy of a number and .05 of the 
aumber is 20. What is the number? 

25. A bushel of corn and a bushel of wheat cost together 
61.50. The corn. costs ¢ as much as the wheat. Find the cost 
of each. 

26. A’s age is # of B’s age. The sum of their ages is 48 
years. Find the age of each. 

27. One-half of the time past midnight equals the time till 
100n. What time is it? 





Hint: Let x = the number of hours past midnight. 
Then, 5 = the number of hours till noon, 
and +5 = 12. 


{ 
| 28. One-third of the time till midnight equals the time 
vast noon. What time is it? 


84 EQUATIONS AND PROBLEMS LCuap. IX. 


29. Three-fifths of the time past midnight equals the time 
till noon. What time is it? | | 

30. Twice a certain number is 7 more than { of it. Find 
the number. 

31. If a certain number is diminished by 99 the result is the 
same as if it is divided by 10. What is the number? 1 

32. The sum of the three angles of a triangle is 180°. In 
a triangle the second angle is 3 times the first, and the third is 
1 the first. How many degrees in each angle? 

33. In aright triangle one acute angle is 3 as large as the 
other. How many degrees in each? 

34. In a right triangle one acute angle is 12° greater than 
the other. How many degrees in each? 

35. In a triangle the second angle is 3 the first, and the 
third is twice the second. How many degrees in each? 

36. At what time between 3 and 4 o’clock are the hands of 
a clock together? 


SuacEstion: Let x = the number of minutes past 3 o’clock when 
the hands are together. Since the hour hand travels 7; as fast as the 


minute hand, 6 is the number of minute spaces over which the hour hand 


has passed since 3 o’clock. But at 3 o’clock the hour hand was 15 minute 
spaces ahead of the minute hand. Hence, : 


37. At what time between 5 and 6 o’clock are the hands of 
a clock together? | 

38. At what time between 2 and 3 o’clock are the hands of 
a clock pointing in opposite directions? 

39. At what time between 7 and 8 o’clock are the hands of 
a clock at right angles to each other? Two answers. 

40. If it takes the author 3 times as long to make up a 
problem as it does the student to solve it, and the making and 
solving together take 12 minutes, how much time would be 
saved each by omitting the problem? 





‘Arr. 60] PROBLEMS 85 


41. In a recent election 46 less than two-fifths of the votes 
were cast by women. Three-sevenths of the women voted a 
certain ticket, thus casting 540 votes. How many men voted? 

42. A man travels for 25 hours in his automobile. At the 
end of that time something happens to the engine, and he has 
to travel at half his former speed for 13 hours to reach a garage. 
His speedometer shows that he has traveled 65 miles. What 
was his speed at first? 

_ 48. The total operating revenue of the Pennsylvania Rail- 
road Company for a certain year was $6,000,000 more than 
4 of the total operating expenses. The total operating revenue 
was $157,000,000. Find the total operating expenses. 

_ 44, A man has a certain sum of money invested at 7%, 
and = as much at 6%. The income from the two investments 
is $248. Find the amount of each. | 





CHAPTER X 


DIVISION 


61. Division of monomials. Since a?- a? =a', it follows 
from the definition of division, Art. 32, that 


a’ +a? = a7, 
Similarly, since 
q™"-.qr = a", 
then, 


ms pn — pm-n 
(Ge Fa Ss fe ‘ 


We have then the following rule of exponents for 
division: 

The exponent of a letter in the quotient equals the exponent of 
that letter in the dividend minus its exponent in the divisor. 

In all cases the quotient must be given the proper sign 
according to the rule given in Art. 32. 





To multiply 3ry? by 42%y2, we form the products of the ‘numerical 
coefficients and then the products of the letters. Hence, to divide 12z%y* 
by 3xy?, we divide 12 by 3, then x* by a, then y’ by y’, and obtain 


12z°y' 
oxy’ 





=47°/", 


If there are factors in the divisor that are not in the dividend, the 
result is left in the form of a fraction. Thus, 
18-23), 2mink Sie 
12 2’ 6mnz 3mz 
86 


‘Ars. 61, 62] DIVISION OF A POLYNOMIAL 


Complete the following indicated divisions, and check the 


EXERCISES 


results by multiplication : 


























3h2 a3 is 
Fe QD 14: 120%? 21. (=2) (-y), 
3ab x 
= 87)? | aby _p)2 
AD. 6 12. —lbmin* 22. (~a) (—6)? 
—5m3 —b 
’ Sxy (a + 1)% 
3. 18a + 3a. 13. ce 23. (a +12 
0.52%y (x — y)? 
rteapel Saati Sa 
4. 30a? + —2. 14. WES 24. (Ga) 
= pet eR) eh 23 + 32-5 
5. TYe + @. 15. Bis ore 25. er a 
ae ase (—m) (—n) (—p)? 
2 6 7 ° 
6. 3a? + 3b. 16. yp 8B. 
Sr yi2" (a — 3) (b + 4)? 
37, 2 m2 ; ° 
7. xty + 2°, 1. ee 8 G3 O ED 
4a*b*c? 122°y724 
ate. 2 - ° 
8. 3a? + 2a. 18. he 28. a 
oe 12p°q —(loxy2 
2; a 6 +5. 19. zy ae 29. 3022” 
— of2 Myn 
Pei 13 211. 20, —. a6 
— 3st art 


62. Division of a polynomial by a monomial. Since 
a(b+c+d) =ab+ac + ad, 

t follows from the definition of division that 
(ab+ac+ad)+a=b+c+d 


' This illustrates the rule : 
To dwide a polynomial by a monomial, divide each term of 
he polynomial by the monomial and add the quotients. 





88 


Divide and check the results by multiplication: 


rs 


ee & 
on & Oo 


i 
ie 


16. 


17. 


18. 


20. 


7.908 


22. 


23. 


OE OC Oe A ae ee 


DIVISION 


EXERCISES — 


3a? + 4a by =z. 

10y — 15a by 5. 

6x? + 8x by 2z. 

ay + 4ay by wy. 

14abx — 49bcex + 7bx by 7b. 

2a — 6b + 10c by 2. 

ge — x? + a4 by &. 

m3 — 2m@x + ma* by —m. 

48x3y2z — 18a°y?2 + 120xy?z* by Oxyz. 
66min? — 1.2m?n by .2mn. 


_ 25a7b%e> — .5a>dbtc? + 2.5a°b?c? by .25a7b*c?. 
. 6a+2z2+ (a+2) bya+e. 
. 12(m — n)3 — 9(m — n)? + 6(m — n) by 3(m — n). 


4a? — 6a + 2 
gc aU ? 15. ie 
a —b —ce ad —A 
—1 
3p? — 6p’g + 38pq" _ » 
Spy 
Alot. di AC a 
elias dae ee Siqcus =? 19 3 
36(a — 1)? + 12(a — 1) aes 
12(a — 1) ; 
18a3y? — 302x7y 
bry 
18(a — 1)? — 30(a — 1)? sis 
6(a — 1) 
98(a — 5)? + 35(a — 5)? — 7(a — 5) es 
7(a — 5) 


=f, 


ae 


[Cuap. X. 


et 7? — 9 


9a + 664+ 3 ee 








IN 


it 
My 


‘Arts. 62,63] DIVISION BY A POLYNOMIAL 89 


24. seen | 2°" ay, 
a?b3 

4 oa 
ab” 


26. Select monomial divisors for the following and divide: 
(zy — 1447+ 7x, 3a + 6a? — 9a’, 8xy + 42°y — 16zy’. 


63. Division by a polynomial. 
Example 1. Divide x? + 2z?y + 2xy? + y’ by x? + ry + y?. 
SOLUTION: 
B+ 2ey+2ey+y wt+ayt+y 
t w+ xyt+ xy? a+y 
| ey+ ry +y 
vy + pty? 


me CHECK: Let z=1, and y=1. Then 2° + 2a’%y +22y? +73 = 6, 
e’+azy+y =3,andzx+y=2. Since 6 + 3 = 2, the solution checks. 


Explanation. (1) It is convenient to arrange the dividend 
and divisor according to the descending powers of x. 

(2) The highest power of x in the dividend divided by the 
highest power of x in the divisor gives the highest power of x 
in the quotient. Dividing x’ by x”, we get x for the first term 
of the quotient. 

(3) Since the dividend is the product of the divisor and 

quotient, it contains the product of the divisor and each term 
of the quotient. Hence, we multiply 2? + x2y + y? by x and 
subtract the product, 2 + xy + xy’, from the dividend. The 
‘remainder, xy + xy’ + y’, contains the product of the divisor 
and the remaining terms of the quotient. 
(4) The first term of the remainder, «?y, divided by the 
first term of the divisor, x?, gives the second term of the quo- 
tient, y. Multiplying the divisor by y and subtracting, the 
‘remainder is zero. The division is therefore completed, and 
the quotient is x + y. 





90 DIVISION [Cuap. X; 


Example 2. Divide 1225 — 262‘ — 152’ + 8a? - 4x + 9 by 423 — 277 + | 
x-l. 


SOLUTION: 


1225 — 262! — 1523 + 82? —4274+9 Mo? — 22? +27-1 
124 — 67° = Se 52" Ba? —5r — 7 
— 20x4 — 182? + 11x? — 42x 
— 2027! + 102° — 5a* + 52 
—2873 + 162? —- 97 + 9 
—2823 + 142? —724+7 


2a? — 2a + 2=Remainder 


EXERCISES 


Divide and check the results by substituting numbers for 
the letters: 


. a+ 2ab+ 0? by a+ b. 
—2ry +y? by x — y. 

e+ 54+6byr+3. 

m? — 7m +12 by m — 4. 

y? — y — 20 by y — 5. 

a + 2a — 35 by a+ 7. 

v’+9byx+4+3. 

. 2+ bax + 8a? by x + 2a. 

a? — Jab + 146? by a-— 2b. 

6a? + Tam + 2m? by 2a + m. 

. 202? — Try — 3y? by 5x — dy. 

a —bya— b. 

. 4a? — 25y* by 2x + dy. 

. & + 3b + 3ab? + B® by a + b. 

. m+ 5m?n — 24n? by m — 3n. 

. 4a? + 4a? — 29a — 21 by 2a — 3. 

. 8a? + 12a%b + Gab? + b3 by 2a + b. 

. 12u3 — 23u%v + bur? + 5v3 by 4u — 5v. 

a — bby a — b. 

} om? + 8n3 by 3m + 2n. 


SO ON Sd ee ae aoe gies 


So ee ee 
SODNARTAPEWNHH OS 








lar. 63] EXERCISES 91 


21. x*—y* by x — y. 

22. v+ytbyxz+y. 

23. p? + g? by pt + qt. 

24. 6x* — 1l3az* + 13a’2? — 13a*x — 5a‘ by 2x? — 3ax — a?. 

25. 4y° — 26y* — 9y*? + 41y? + 2y — 12 by 4y? + 2y — 3. 

26. 21a* — 16a*b — 5ab? + 16a7b? + 2b4 by 3a? — ab + B®. 

27. 4m°n — 4m'n? + 4m?n*t — mn® by 2m’ — 2mn + n’. 

28. a? + 6% — meg ee ore 0: 

29. <2? — tee —iby 3 aE +4, 

30. 8x3 + 27 by 32 4+ 2. 

31. gya* — ayo'y + rery? —gy y® by 32 — ZY. 

32. 4m4 — 2m? + 23m? + 3m + qe by Fm? — 3m — 

33. .16m‘4n? — .01p*q* by .4m?n + .1p%¢’. 

34. .0403 — .12y° + .17xy? — .12%*y by .2% — By. 

35. -.002482° + 1 by .32 + 1. 

36. 1 by 1 + 2 to five terms of the quotient. 

37. 1 by 1 — x to five terms of the quotient. 

38. 22" + Qarry” + y?" by x” + y". 

39. a3" + 3a2"b" + 3a"b?” + 6" by a” + b*. 

40, a3rt3 _ yont9 Hy grtt — y2nts, 

41. Show that ‘pata baat 

5 AE os | 

42. Show that. 2m* — 5m'n + 6m?n? — 4mn? + n* divided 
by m? — mn + n? equals 2m? — 3mn + n? when m = 1, and n=1. 
43. Show that a® — b® divided by a — b is equal to at + a*b 
+b? + ab? + bt when a = 1, and b = 2. 

44. The area of a rectangle remains 1 while the base and 
| change. Find the corresponding values of the base 
‘when the altitude has the values 3, 2, 1, .1, .01, .001, .0001. 
Which of these rectangles has the fangs verter? Which 
‘has the greatest peuetr! 


: 45. In the fraction + — let 2 take on the values 1, 2, 4, 8, 16, 


P| 


=2x+ywhenz = 1, and y =2. 


32, 64, and soon. So ae as possible, represent the correspond- 


92 DIVISION [Cuap. X. 


ing values of the fraction on the number scale. What value 
is approached by the fraction as © becomes larger ? 


46. Find the corresponding values of the fraction ca 
when x takes on the values 1, 2, 3, 4, and so on. So far as | 
possible represent the values of the fraction on the number 
scale. Find a value of x that will make the value of the frac- 
tion greater than .999. What value is approached by the 


fraction as x becomes larger? 


64. Literal coefficients. 





Subtract : 
vk; ax 6. av Lion 
ba Sie. —Nx 
(a — b)x 
2. cy 7. —62 12. ap + bq 
—dy —M2z 5p — mq 
(c+ d)y 
3. ax Ser 13. 32? + bx 
: —bx Ss ka? — 9a 
(a + b)x 
4. am 9. 8xy 14. nr? — mr 
bm axy 1 at 
5. 3m 10. ar 15. ar —by +c 
am Mie —2x — dy —2 


In each of the following expressions the terms have a com- 
mon factor. Find this common factor and write each ex- 
pression as a product. 

16. ax + ay — az. 

SoLturion: The common factor is a. Then, 


ax + ay —az=a(r+y—2). 


it 
i 
| 


i 


ker. 64] EXERCISES 93 


17. bx + by. 24. a*y + 2aby + b’y.. 
18. 6x — by. 25. x — 3x + an. 

19. a? — 3a. 26. 15a — 20ab + 5a. 
20. p+ prt. 27. aba? + aby — ab. 

21. y — 3y. : 28. amx + abm — 4amy. 
22. 7n — 14m + 35k. 29. 2x — by +2. 

23. nr — dr? + Sr, 30. ax — bx + cx — &. 


The following exercises illustrate the use of the division of 

yolynomials in the solution of equations. 
Solve and check results: 
31. ax — a’ — 3ab = 2b? — bre. 
SOLUTION: ax — a? — 3ab = 2b? — br. 

ax + bx = a? + 3ab + 2b?. 

(a + b)x = a? + 3ab + 207. 

_ @ + 3ab + 20? 


a+b 
= a-+ 2b. 


Cueck: Substituting a + 2b for x in the given equation, we have, 
a(a + 2b) — a? — 3ab = 2b? — b(a + 2b). 
Multiplying, . a? + 2ab — a? — 3ab = 2b? — ab — 20’. 
Jollecting terms, —ab = —ab. 
32. 2x + 3ax = 10a + 15a?. 
88. ax + bx = c(a + bd). : 
| 34. ax — bx = a? — 3ab + 26°. 
$5. ax + bx = c(a + b)(c + a). 
(36. ax + 2ab — & = a? + ba. 
87. ax +4=a8+ 30+ 3a+1. 
38. ax — bx = a’ — B. 
39. mz +3m=2+m? + 2. 
40. x +1 = 267+ 5b — be. 


a 


a DIVISION . [Cuar. 38] 


REVIEW EXERCISES AND PROBLEMS 

14. Find the sums of (a) 62, —2x, -32; (b) 5(m+n), —(m + ai 
—8(m.+n) ; (6)/ 73,3, lle, (d) an, 38n, —bn; (e) zy, y, —cy- 

2. Arrange 622 —« + 9x5 — x! +5 according to ascending powers 
of x. : 
3. Arrange a> + 4ab? + ab — a‘b4 + 6a*b? according to descending - 
powers of a; of b. 

4. In each of the following pairs of numbers tell which number is_ 
the greater, and which has the greater absolute value: (a) —4, 7; -(b) —9 
—1; (c) -6,0; (d) —5,5 

5. State a rule for adding like terms. 

6. State a rule for removing parentheses; (a) when preceded by the’ 
sign +; (b) when preceded by the sign —. 

7. State the rule for combining exponents in finding the produ 
am-aqn-ar. 

Complete the following indicated multiplications: 


8. m3-m™-m. 12. (3)?-4. 16. 6(27—4y + 3). 

9. 2a3xy - Saxy2?. 13..(— 2)" (=a. 17. sx( 20 aed 5): 
405 

10. (a3)? - (ab)*. 14. 3ab- 4a. 18. 3(2-3 +4-3? — 3%)+ 


: ae 
11. (—a4)8- (2ax)*. 15. mn. 


‘19. State the rule of exponents in dividing am by an. 


Complete the following indicated divisions: 








20. 12a%ty'+4atmy’. 24. (2a+2d) +2 28. aot 
Q1. (28-4) +2. 25. 3m + 3mn 29 bee jive 
3m ap ae | 
6 +14 (a — b)? 

92. (23 +4) +2. 26. : eter 

Lat ee) 6 349 ent i 
23. (2a-+2b) +2. Teese 91, Sa 

5 2a 
30. asb - ab? 

ab 


Zs Simplify: « — 3y — (@ +3y) - {1+ [e@+y -@+2y —2) +4] 
— 72x}. 


Arr. 64] REVIEW EXERCISES AND PROBLEMS 95 


34. Simplify: 2a + (3b -a +4) — [2 + (3b —4) —(1 +a). 

35. Multiply 500+70+5 by 200 +30+4, using the form for 
multiplying one polynominal by another. Compare this result with that 
obtained by multiplying 575 by 234 in the usual way. 


36. Compare in a similar way the products (42 + 3)(18 +3) and 
42? X 18%. 
Solve the following equations for z: 
Bier 44 3 — 1) = —2(x — 2). 
38. 27 — a= 3x +b. 
39. az + 0? = br + a’. 


40. 


™ 41. mz — 3n = 3m — nz. 
| ‘42. In the equation 2x + b = 3, give b numerical values such that 
she value of x shall be (1) a positive integer; (2) a negative integer ; 
3) a positive fraction ; (4) a negative fraction ; (5) zero. Which of these 
values of x have no meaning if x is the number of points scored in a foot- 
all game? 

43. Find six terms in the quotient 1 + (1 —). What is the difference 


»etween the sum of these six terms and i : = when « = 4? Whenz=.1? 


When x = .01? 
44. Show that the numbers of the series .3, .33, .833, .3333, and so on 


proach nearer and nearer the value }. 





45. If A is the area of a square whose side is s, then A = s?._ By what 

3A multiplied if s is multiplied by 2? If s is multiplied by 3? By 4? 
3y 10? By a? 

46. 3 t= 10, what number is represented by 3¢4 + 52 + 8¢ +5? 

ope 0, 

Wat) ess ay! 
_ 47. Write 86423 and 23090 in terms of powers of ¢ = 10, and sub- 
‘ract 5 times the second from the first. Check for ¢ = 10. 


48. Write 67731 and 211 in terms of powers of ¢t = 10, and divide the 
: rst by the second. Check for ¢ = 10. 


CHAPTER XI 
LINEAR EQUATIONS 


65. Linear equations. A great many of the equations 
we have solved can be reduced by the use of the four fundamental 
operations to the form : 

ax +6 =0, 
where x represents the unknown, and a and 6 are numbers 
that may have any value except that a cannot be zero. 


Thus, 42 +3 =a —7 can be reduced to 3x + 10 = 0, which is of the 
above form, where a = 3,b=10. Again, iy +2 =3(1 + dy) can be put 
in the form ay +b = 0, where a = 3,6 = -1 


Equations which can be reduced to the form ax + b = 0 by 
use of the principles given in Art. 38, are called linear equations 
in one unknown. Such equations are often called simple equa- 
tions. 

EXERCISES AND PROBLEMS 


Write the following equations in the form az + b = 0 and 
point out the values of a and 6: | 


1.. 8¢ + 2'= 62-4 6. 

2. 62 — 5 = 92 + 2. 

8. 5¢4 —3 + 21r7 = 18 + 42. 
4. 3(2 —7) =“ +415. 

5. 81 =—4(e 4+) =—2-7 
6. 25 — 6(x — 6) = 20 — (2% — 13). 
7. 13(2% — 1) = 5(52 + 4). 

8 





a, aes 
secre 3 
7 —3(@-—5) | 
9. m = |], 


96 





i 
| 
i 
if 
| 


Arr. 65] EXERCISES 97 





y 


10. x - 2-34 = WGr +1). 
E Hint: Multiply each member by 36. 
Which of the following equations can be reduced to the 
type form ax + 6 = 0? 
11. x(x —1) = 404 + 3. 


SotuTIon: Collect terms and this equation reduces to 2? — 5a — 3 =0. 
‘Since it contains a term in 2? it is not of the form ax +b = 0. 


m 12. (x + 3)(a — 2) = 4(a@ — 2). 

‘— 13. v—44+5=54 2", 

my 14. (27 — 1)(82 + 1) = (62 — 12)(@ + 3). 

1b. (x — 5)(27 — 9) = (& — 4) (22 — 6). 

fe 16. x(x? — 1) = 242 + 3. 

m 17. (22 — 1)(144e + 5) — 267 = 367(8% + 1) +11. 


; x AP x 


x d5xe+4+1 
19. (x — 2)@ +2) +3 = iG + 30. 


20. (« + 1)? + ( — 2)? = («@ —1)(@#+ 5) + 2. 








Solve the following equations : 
21. 9x + 5x + 21 — 68 — 6 = —22 — 5B. 
92. 3° 5(x + 6) + 5° 7(1 + 2x) — 7: O(x — 8) = 827. 





23. { — 5a = @ — 28a. 
24. i -etz- G7 18. 
25. 8e —5 = +5 + 153 
26. a(x - §) - (5 +5) 
7. S+5t1-5-5 
98. ett ts 





98 LINEAR EQUATIONS [Cuar. XL 


29. +32 =3+4 92. 

30. 4a + 5a = 5(a@ +). 

31, (2 eo) eee 

$2. (x — 2)? — (4 —3)2 =@. 
$3. 2+a2+1=(¢+4+1). 
34. 244242 = (& + 2). 


oe x 
35. f +20 = (5 +1): 
36. (e+ a)(@ — a) = 2 + 2x + 2a’. 


87. (~@+ nS oa 2) = (*% + 2(5 + 1); 


38. 922 — 1 = 3(82? + 52). 

39. (2x — 5)(27 + 5) = (4r — 11)(a@ + 1). 

40. When 4 is subtracted from twice a number the result 
is 30. What is the number? 

41. When 4 is subtracted from a number and the result 
doubled we get 30. What is the number? 

42. The sum of two consecutive integers is 27. What are 
the numbers? 

43. The sum of three consecutive integers is 27. What 
are the numbers? : 

44. Is it possible to find 4 consecutive integers whose sum 
is 27? 

45. Find 4 consecutive integers whose sum is 46. 

46. Find three consecutive odd integers whose sum is 39. 

47. Find three consecutive even integers whose sum is 42. — 

48. A teamster contracts to haul 2100 bags of flour. He 
makes 22 trips with his dray carrying the same number of 
bags each trip except the last one when he carries 42 bags, 
How many bags does he haul each of the first 22 trips? 
; 49. The perimeter of a rectangle whose length is 4 feet 

longer than its width is 28. Find the dimensions. 

50. The length of a rectangle is 3 feet more than twice the 

width. The perimeter is 42. Find the dimensions. ~ 





(Arr. 65] PROBLEMS 99 


51. A cubic foot of pure water weighs 62.5 pounds. Twenty 
cubic feet of water weigh 15 pounds less than 22 cubic feet 
of ice. What is the weight of a cubic foot of ice? 

__ 62. One pound of ice occupies 30 cubic inches of space. 
Five pounds of ice when melted decrease 12 cubic inches in 
volume. What volume does 1 pound of water occupy? 

53. EKighty gallons of water from the Dead Sea weigh 
1 pound less than 100 gallons of pure water. The weight of a 
gallon of pure water is 8.33 pounds. What is the weight of a 
gallon of water from the Dead Sea? 

I 54. A father 54 years old has a son 21 years old. How 
any years ago was the father 4 times as old as the son? 

' 65. What number is to be subtracted from both the nu- 
‘merator and denominator of 3+ in order that the new fraction 
may be equal to 4? 

56. The denominator of a fraction is 12. When 3 is sub- 
tracted from both numerator and denominator the value of 
Mie fraction is decreased by 34. What is the numerator of 
the first fraction? 

57. A number is the sum of two parts. The first is 3 
‘greater than half the number, while the second is 2 greater 
than one-fourth of the number. What is the number and how 
was it divided? 

_ 68. What number has the property that when multiplied 
by & the result is greater by one than when multiplied by #? 











CHAPTER XII 


IMPORTANT TYPE PRODUCTS 


— 


66. Certain algebraic products occur so frequently and | 
are so useful as models for other multiplications that they | 
should be memorized. | 


67%. Square of a binominal. Multiplying a +b by a + ‘a | 
we find | 
(a+b)? = a@ + 2ab re b?, 
~ which may be translated into words as follows : | 
The square of the sum of two numbers is the square of the | 
first, plus twice their product, plus the square of the second. | 
In a similar way, we find 


(a — b)? = @ — 2ab + OB, 
or in words : 

The square of the difference of two numbers is the square a | 
the first, minus twice their product, plus the square of ie | 
second. | 

In these formulas, a eit b represent any two numbers, or 


any two expressions. 
Example 1. (10 +5)? = 10? +2:-10°5 + 5 = 225. 
Here a = 10, and b = 5. 
Example 2. [ay? + (x + y)]? = (xy)? + 2(@zy)@ +y) + (@ +9). 
Here a= ary’, andb=2x+y. 


An algebraic expression which is the product of two equal 


factors is called a perfect square. 
100 








. 


‘Arr. 67] EXERCISES 101 
; EXERCISES 
Square the following binomials by the above rules: 


1 ¢v7+/y. 4. “2+¢. 7 1+2. 
5. 24.3. 8. 7+ 5. 
6. 2 — 3. 9. 646. 





Fia. 17 


10. Let ABCD (Fig. 16) be a square whose side is of length 
\a+6. Its area is then (a+b). Let AEFG be a square of 
side a. From the figure show that (a + 6b)? = a? + 2ab + b?. 
) 11. Let TUVW (Fig. 17) be a square whose side isa. Let 
|) XU =b. Then TXYZ is a square whose side is (a — b). 
(From the figure show (a — b)? = a? — 2ab + B*. 
Square the following numbers by expressing each number 
_as the sum or difference of two other numbers and then applying 
‘the above rules. Work each exercise in two different ways. 
12. 39. 
SOLUTION: 39? = (30 +9)? = 30? + 2: 30-9 + 9? = 1521. 

39? = (40 — 1)? = 40? —-2- 40-14 12 = 1521. 


Sees Oy. 15. 1, 16. 0. 17.51. 18. 99. 


| Square the following according to the above rules and 
‘verify by actual multiplication : 

H 19. a? — y. 21. ry — 2. 23. cy +2. 

20. x + 3y. 22. ab + xy. : 24. 2a + 32. 


102 IMPORTANT TYPE PRODUCTS  [Cuav. XIL. | 


25. xy — y’. 28. (a+b) +6. 31. x+y - 2. 
26. 5k — 3h. 29. 999. | 32. m—n—?. 
27. a+ 110. 30. r+y+2. 33. 2h +k — 3l. 


The following are squares of binomials; find the two equal | 
binomial factors: | 


34. 2 4+2cd+a@. 38. 4a? + 12ab + 9b. 
35. 2? — Qry + y’. 39. 922+ 6241. 

36. 7? — 427 + 4. 40. 252? + 20x + 4. 
37. y? + by + 9. 41. 49r? — 42r + 9. 


- 42. Give a rule for finding whether or not’a trinomial is a 
perfect square. : | 


Hint: Two terms of the trinomial must be perfect squares. What is | 
the other term? 


Some of the following are perfect squares; find them and | 
give the factors: 





43. c+ 2cd — d’. 48. 9 + 42 + 49. 

44, x? — 102 + 25. 49. 4-54 25. 

45. 16a%y? — 8ry + 1. 50. 49a?y* + 1l4ay? + 1. 
2 

46. 9a2 + 7a +1. 61. Gtue+4. 

47. vy?+ay +1. 52. + B48 


68. Product of the sum and difference of two numbers. 
By carrying out the multiplication, we find 


(a+b) (a — b).= @ — BW, 
In words, this formula reads : 


The product of the sum and difference of two ee: 1s the 
difference of the squares of the two numbers. 


i 
' 
“Arr. 68] EXERCISES 103 
i 

EXERCISES 


Form the following products by the above rule and verify 
by actual multiplication : 


1. (vx —y) (@+ y). 7. (3a + 2b) (8a — 2b). 
2. (c +d) (c —‘d). 8. (x — y") (4 +»). 

3. (x +2) (x — 2). GF (le ey (1 42) 

4. (3x + y) (32 — y). 10.8010 1) (10 4-1), 
5. («@ — y’) (@+ y’). 11. (a? +3) (@ — 8). 

6. (2 +2y)(-r+2y), 12 (+2) (-2). 


| The following binomials are the products of the sum and 
Wiifierence of the same two numbers. Find the numbers. 


13. a? — 4b. 15. 16 — 2". 17. o**. — 7! 
14. 42? — 9y’. 16. 9 — 25. 18. xt — y'. 


Perform the indicated divisions : 

19. (2? — y*) + (@+ y). 
/ 20. (at — b*) + (a? — bd). 

51. (s? — Or?) + (s — 3r). 

22. (16n? — 36m?) + (4n — 6m). 

23. (a? + 2ab+ 0? —c’?) + (a+b-4+0). 

24. (x? — Qry +.y? — 92?) + (x — y + 82). 
25. Give arule for telling whether or not a binomial 
is the product of the sum and difference of the same two 
numbers. 


Which of the following are the products of a sum and dif- 
ference? Find the factors of those which are such products. 
26. (a — b)? — ce’. 29. a +0? — c?. 
a7. at +B. 30. a2 +a +1)- 
Wes. 84 2241-42 81. a + 2041-2 





104 IMPORTANT TYPE PRODUCTS  ([(Caap. XID 


69. Product of two binomials having acommonterm. ‘l'wo 
binomials having a common term can. be written in the form 
z+aand«#+b. By actual multiplication, we find 


re+a 

a+b 

x + ax 

ba + ab 

x? + (a+ b)x + ab 

or (a +a) (wv +b) =a? + (a+ b)x + ab. 
In words this reads : 

| The product of two binomials having a common term equals 
the square of the common term plus the product of the common 
term by the sum of the other terms, plus the product of the other 
terms. 


EXERCISES 
Expand by the above formula : 
1, (2 +2) (« +3). 4. (x —1) (&@ — 2). 
2. (a+ 2) (a +3). 5. (a — 4) (a + 5). 
3. (x + 5) (@ +2). 6. (n — 3) (1 — 39). 


T.. (22 + 3) (2x +1). 
SoLuTIon: (22 + 3)(2a 4+ 1) = (2x)? + (8 +1)2a4 +3°1 


= 4x? + 8x +3. 
8. (3y.+ 1) (8y + 2). 10. (4 + 3a) (4 — 2a). 
9. (2a +b) (2a +c). 11. (av + 13) (ax — 1). 


The following trinomials are products of two binomials 
having a common term; find the binomials: 


12. 2° + 37 +2 = (x +2) ( ). 18. y+y-—6. 


13. 2? + 42 + 3. 19. a? — 2ab — pos 
14. 2+ 67+ 5. 20. 25+ 45+ 14. 
15. n?+ 7n + 10. 21. w? — 12u + 32: 


16. a? —6a4+ 9. 22. (x +1)? + 8(@+4+1) -4. 
17. «x? — 3x + 2. 





i 


| 
i 


‘Arrs. 69, 70] | MISCELLANEOUS EXERCISES 105 


MISCELLANEOUS EXERCISES 


Form the following products according to the foregoing 


rules: 


1. a _ v) G + v), 9. 
(2c +1) (x + 4). 10. 
(x +3) (x + 3). it: 


. (2P + 4) (2P + 6). 


edie bite Ls: 


(xy + 2) (ay — 2). io: 
(z+ 6) (+7). 14. 
(x — 1) (x + 2). 15. 
(a — 4) (a +4). 16. 


(ab — xy) (ab + xy). 


(x +3) (x — 16). 
Ge 9) a — 47°): 


(1 +c) (1-2). 

(9r242 + 1) (9r?? — 1). 
(a+b) +c] [@ +b) —c]. 
(x?y? + 7) (xy? — 2). 


The following expressions are either the products of the sum 
nd difference of the same two numbers, or products of two 
binomials having a common term; ss the factors of each : 


LT. 
18. 
19. 
20. 
21. 


70. 


x? + 5a 4+ 4. 

a? + 7a + 12. 

100 — 9a?. 

a? + 4a +4 4. 

(x? — Qay + y”) — 2. 


22. — 9x + 20. 

23; + 2ab + b? — s? + 4s — 4. 
24. p> + 7pq + 104°. 

25. 16 + 20x + 627. 


Cube of a binomial. By actual multiplication, we find 


(a+b)? = (a+b) (a+b) (a+b) =a+6 


a+b 
a? + ab 
ab + 0? 
a? + 2ab + 6? 
a+b 

a? + 2a2b + ab? 

a’b + 2ab? + 6 
a? + 3a7b + 3ab? + 63. 








106 IMPORTANT TYPE PRODUCTS — [Cuap. XII. — 


That is, the cube of the sum of two numbers a and 6 consists — 
of four terms as follows : a 


The first term = a 


the cube of the first number. 


The second term = 3a*b = three times the square of the first 
multiplied by the second. 


_ The third term = 3ab’ = three times the first multiplied by 
the square of the second. 


The fourth term = b? = the cube of the second. 
As a formula, we have 


(a+b) = @ + 3a°b + 3ab* + BD. 
In a similar way, we find | 
(a — b)? = & — 3a’b + Sab * — B. 


EXERCISES 
Expand the following by the foregoing rule or formula : 
1. (c + d)°. 4. (« + 1). 7. (3a — b)3, 
2: (27 —y)*. 5. (2 + 3)3. 8. (52 —2)*. 
3. (a + 2)%. 6. (2x + y)’. 9. (xy + 1)%. 


10. (xy + 22). 


71. Square of a trinomial. By actual multiplication, we 
find 
(at+b+eP=040? 4c? + 2ab + Zac + 2be. 


That is, the square of a trinomial equals the sum of the squares 
of its terms plus twice the product of each term by each succeeding 
term. ; 


EXERCISES 
1. By actual multiplication prove: 
(a+b—cP=0+h+4c + 2ab — 2ac — 2be. 
2. Prove (a—b—c)? =@+0?+¢ — 2ab — 2ac + 2be. 
8. Prove (a—b+c)=@+4+0?+c? — 2ab + 2ac — 2bc. 





PQ Ta ar See 


‘Arr. 71] | EXERCISES 107 


By use of the foregoing formulas expand : 


4. (x+y+z)?. 8 (142+ 3)?. 

mo. (c— y + 1)?. 9. (1+1+1)?. 
6. (a + 2b + 3c)’. 10. [(a +b) +2+y]?. 
q. (ar — y + 32)". AT eo 0 ae), 


MISCELLANEOUS EXERCISES 


Perform the following multiplications and divisions, using 
the type products of this chapter whenever possible. Check 
the results by substituting special numerical values. 





1. (14+ a)? 14.. (1 +a + 2a)’. 
2. (1 +n) (1 + 2n). 15. (402+ 44 + 1) + (Q4 41). 
les. (1 —7n) (1+ 2n). 16. (2? — 32 + 2) + (x — 2). 
24. (1 —n) (1 — 2n). 17. (a-— 4) (a+ 3). 

5. (1 +n) (1 — 2n). 18. (a + 4) (a — 3). 

6. (1 — 3a) (1+ 3a). ~ 19. (a+ 4) (a+ 3). 
m7. (1 + 2)2. 20. (a — 3) (a — 3). 
eo. (1 + a)*. ate AL 

2 4 — 16¢ 

9. (2x — 3)?. 22. Ti pSaae 

ett — 2 — y) (1—2z +4). 
11. (xy — 8) (xy + 1). 23. (101) (99). 

2. (ct+y—8) (xt+ytl). 24. (a? + 262 +169) + (r+ 18). 
13. (1 + 2)%. 25. (a? + ax — 2a”) + (a + 22). 
| b 

| 
| | 
Fig. 18 K Fia. 19 


26. What are the dimensions of the rectangles in Fig. 18? 
Show that the two rectangles can be joined together so as to 


108 IMPORTANT TYPE PRODUCTS [Cuar. XI. 


make a rectangle whose length is a + 6, and whose width is 
a—b. From the figure show that (a + b) (a — b) = a — B’. 


27. Fig. 19 is a rectangle of width x + a and height x + 0b. 
From the figure show that (v + a) (v + b) = a 4+ (a+ b)x + ab. 
Fill out the blanks in the following : 
28. ab? — 64c2d? = (ab + 8cd)( ). 
29. a3 + 3a2x + 3az? + 23 = (a? + Zax + x*)( ). 
30. 44:27 4 2y + 2y = 2 Paya 
31. (xy? — Qey +1) = (zy — 1)( ). 
32. (2x +a) (24 +b) = 427 +( ) +8. 
Remove parentheses and unite terms where possible : 
33. (2% — 3)? + 3(82 — 2). 
34. (xy — 2)? + (ay + 2). 
35. 6a(a — 1) + 2(8a — a)?. 
36. 62(a — 1) + 2(a — 3z)?. 
37. a(b — a) — (2 — a?): 
38. (a? — b?) — (a — ). 
39. 6(2? — 37) — 12(2 — 3) 
40. (5% — 3%) — (5 — 3)’. 
41. (a? = b*) — (a — Bb)’. 
42. Show by multiplying that 
a(a — x)(y + 2) + x(a — y)(a — 2) — 2ayz 
=a(a — y)(@ +2) — y@ —- a)(a — 2) — 2axe. 


CHAPTER XIII 
FACTORING 


72. Prime factors in arithmetic. One of the problems of 
arithmetic is that of finding the factors of a given number. A 
factor is one of two or more numbers whose product is the given 
number. Thus, 30 may be written 2-15, 5-6, 3-10, or 2-3-5. 
The numbers, 2, 3, 5, 6, 10, and 15 are all integral factors of 
30. If we consider fractions we may go on indefinitely, for 
we may write 30 = 7-24, 30 = 4-$-36, 30=4-32-195 and 
sO on. 

The important problem in this connection, however, is find- 
ing the prime factors. A prime factor is an integral factor which 
is the product of no two integers except itself and unity. There 
are many sets of numbers whose product is 30, but there is 
only one set of prime factors, that is, 2,3, and 5. Any integer 
has only one set of prime factors. 

_ By the factors of a number we shall often mean its prime 
factors although the word prime is omitted. 


EXERCISES 


Separate the following numbers into products of prime 
factors; find for each number two.sets of factors which are not 
prime: 


i 12. Baro. 5. 32. (ER MENE 9.570; 


i 


2. 15. 4, 120. 62-31. 8. 100. 


) 

) | 

_ @3. Prime factors in algebra. The expressions which we 
propose to treat in this chapter involve a definite number of 
Additions, subtractions, multiplications, and divisions, but no 


other operations. Such expressions are said to be rational 
109 


110 FACTORING [Cuap. XIII, 


expressions. A factor of a rational expression is one of two 
or more rational expressions whose product is the given ex- 
pression. 

An expression that, is rational with respect to. a given letter 
contains no indicated root of that letter. 


3 : 
Thus, a, : +=, a? + a are all rational with respect to a and x. The 


expression @ + ax + ./zx is rational with respect to a, but not with respect 
to x. The expression involves, besides the operations of addition and 
multiplication, the operation of extracting the square root. 


An expression is integral with respect to a letter if this letter 
does not occur in any denominator. 


3 
Thus, a? + = is integral with respect to x, but fractional with respect 


to a. 


A factor is rational and integral if it is rational and integral 
with respect to all the letters contained in it. 


Thus, a? + ay + 5a? is rational and integral. 


In this chapter the word factor is to mean rational and 
integral factor. : 


Thus, the factors of 22 — y? are x —y and x+y. Although 2? —¥ 
= V2 — y- Vx? — 2, we shall not consider V2? — y’ as a factor of 2? — y?, 
Again, the factors of } — 2? are  — x and 3 +2; the factors of 2x3 + 22? + 
2x are 2, x, and a7 +2 +41. 


A factor is said to be prime if it contains no factor except 
itself and one. The different prime factors of 15a%b’c are 3, 5, : 
a,b,andc. 15, a’, b?, ab are also factors, but not prime factors. 
In ARGS as in arithmetic, the most important sees of 
factoring is that of finding the prime factors. 


44. Factors of monomials. The factors of a monomial 
are evident by inspection. 
Thus, the different prime factors of 21a?z* are 3, 7, a, 7 % | 


\Anrs. 74, 75] MONOMIAL FACTORS 114 





EXERCISES 
Name the different prime factors of the following : 
mi. Ga*b*. - 4. Qlab?cid*. Dh abecsd?, 
a 2. 12a?y%z. 5. 32p7q°r. 8. 169272". 
(8. 42meniaty. . a. 33 + D2sH5. 9. 1728(abc)?. 


| ! 10. —108(xy?z*)3, 

_ 11. If ab?’ is considered as one factor of 6a2b2, what is the 

other? 

| 12. If 42m?n*xy is considered as the product of two factors 

and one factor is 6mz, what is the other? 

| 13. 32p’@r is the product of three factors. Two of them 

re 2p and pqr. What is the other factor? 

| 14. If a®b’c'd’ is obtained by multiplying together three 

oxpressions, two of which are a’c? and bd, what is the third? 
15. The expression 9lab?c’d! is apraned by multiplying 

‘together three expressions. One of these is 13bcd. Write down 

i; three possible pairs of the other two factors. 


| 75. Monomial factors in polynomials. If each term of a 


dolynomial contains the same monomial factor, then this 
enomial factor is a factor of the polynomial, and the problem 
| s to obtain the other factor. (See Arts. 45, 53, 62.) A type 
‘orm of expression coming under this class is 








| ax+ay—az=a(r+y-— 2), 


17 the factors of ax + ay — az area andz+y —z. 
: Thus, in factoring 6a*b -- L2a*%* — da*b? we find that each 


6a‘b — 124%! — 3a°b? = lon — 4b — b), 


EXERCISES 
Factor the following : 
1. mx — my. 3.00? 4, 
2. 4x — 8. te ad) ee) 





112 FACTORING [Cuar. XI) 
5. 6a? + 16a. 8. 2x + 4y — 8z. 
6. 32 + Ory + 1827. 9. 6a? — 9a? + 3a. . | 
7, a + a°b + ae. 10. 5a + 202? + 152°. 
1 


11. 2x +2y +ax +ay =2e+y)+ae+y) = (2+a)( ), 
12. 4ax + 4ba — 3a — 3b + Say + Sby 
~ 4r(a +b) — 3(a +b) + 5y(at 6) =( )(). 
13.- 3a + 3b + 3c — 2ax — 2bx — 2cx 
-3(a +b +e) —2e(at+b+e) =()(). 
14. Write down a rule for factoring polynomials havi a 
common monomiat factor. 


6. Factors found by grouping terms. As shown in Exer- 
cises 11, 12 and 13 of the last article, a polynomial may have. 
Pe enomial factors which may be found by grouping the terms 
properly. A typical example of this class is aw + ay agl + by. 


Collecting the terms in parentheses, we get ay 
- 


(ax + ay) + (bx + by) =a(e+y)+b@t+y). TF 


Each group of terms contains x + Y, which then may be factored 
out. This gives 


a(a+y) +b(@ +y) = (x + y)(a + 9B). 


2. -- + eke gteeyyepatlinn age oman. 


Hence, a] 
ax + ay + bx + by = (a +y)(a + 0). 1 
) i 

EXERCISES 
Factor the following : | 
1. 2(a +b) +2(a +d). 3. 2(22 + y) + x(2u + y). | 
2. ax +y) + (e+ y). 4. a(b—c)+b(b—c). “@l 


3a(2y — 32) — 4y(2y — 32). 

2(4a — 3c) + 3(4a — 3c). 
Watb+c)+alat+b+c). 

. a(x +y) + y(a ty) +2 +). 


— ee 





CO nae 
aeecealal 


ft 


f 


‘Arts. 76, 77] DIFFERENCE OF SQUARES 113 


Hint: 


9. a(a +b) + 2a(a + b) + b(a +). 
10. n(x — y) + my — 2). 


Write the expression n(x — y) — m(x — y). 


MISCELLANEOUS EXERCISES 


Factor the following : 


Le 


— 3a(b + c) — 4(b +c) + 2(6 + €). 


2. aa+b+c)+b(a+b+c). 


pial Pica et geld sen Aa 


10. 


4a*b — 10bc + 6ab. 

8a? + 403 — 2x4 4+ 62°. 

a(a — b) + 2(b — a). 

a*b — ab? + a*b® — ab‘. 
afa+b+c)+b(a+b+c)+c(a+bd-+0c). 
Avty — 12x4y? — 1l6xty® + 8arty?. 

Write the prime factors of —62527y’. 

Write the prime factors of 27xy(a + y)’. 


_ %%. Difference of two squares. The typical form for this 


ease is 


2 2 
a? — 6, 


which we have seen to be the product of the sum and difference 
of aand b. Hence, 


aw — b= (a+ b)(a— D). 


EXERCISES 
Factor the following : 
i ee 2 8. x* — 16. 
2a°7 — 1. 9. x*— 1. 
3. a — 4. 10. 16a* — 81. 
4, n?— 9. 11. (2a — b)? — 6. 
By te — a: | 12. 25 — 4(x — y)?. 
oe 9 — 4. 18. (a +6)? —-@+y)*: 
7. «4 — y'. 14: (2a — b)? — 9(@ — 1). 


Sotution: (#4 - y’) = (+ yY)@ -y?) =@+y)@+y)@ —y). 


114 FACTORING [Cuap. XIII. 


MISCELLANEOUS EXERCISES 


Factor : | 
1. 3abc + Ya*be. 2. atz* —1. 
3. p(p — g) —a(q — DP) + 29 — P). 
4, atx — dy’. 10. wv — y%. 
5. vt — y’. 11. ab + b& + ac 4+ cz. 
6. 362° — 1. 12. 2° — 1. 
7. 2xryz — 8a*yz — 16zry’z. 13. 17ab’xy? — 34a*b*27y?. 
8. mx + my + nx + ny. 14. 16924 — 7°. 
9. a® — D®. 15. a*x? — a. 


48. Trinomial squares. We have seen that | 


a? + 2ab + b? = (a + b) 
and a2 — 2ab + b? = (a — D) 


from which we see that if in a trinomial the middle term 1s 
twice the product of the square roots of the other two terms, 
then the trinomial is a perfect square; that is, it is a product 
of two equal factors. 

Thus, in xy? — 2xy +1, we have 2ry = 2° /xy?* /1 and 
the trinomial is a perfect square, (ry — 1)’. 


EXERCISES 


Test the following trinomials to see if they are perfect 
squares. If they satisfy the tests, find the two equal factors. 


1. 22+ 2ry 4+ y’. 5. a + 2a+1. 
2. x? — Qry + y’. 6. 2? + 3ry + y’. 
8. v2 —ayt+ y’. 7. 40? + 4ary + 2y?. 
4. v2 + Qry + y’. 8. 44+ 16+ 16. 

9. 70? +2:°70°646. 


“Arr. 78] MISCELLANEOUS EXERCISES 115 


In each of the following expressions replace the parentheses 
“by a term which will make the trinomial a perfect square: 


10. 2?+()+y%. 13. 400+ ( ) + 32. 
a. + ( ) +I. 14. 2? 4+ 42y +(). 
12. 407+ () +a’. 15. 25y?+ ( ) +4. 


16. 169n? + ( ) + 169m?. 


| 17. The middle term of a trinomial is 2xyz. Find three 
possible pairs of terms which will make the trinomial a perfect 
“square. 

Each of the following terms may be considered as the middle 
term of many trinomial squares. Find three such trinomial 
squares for each exercise. 

18. 2abc. 19. 6ryz. 20. 4a%b. 


Separate each of the following into two factors : 


21. 225x?y* + 120x*y* + 162xy’. 
22. 36a°b? + 60a*b® + 25a*b4. 
23. 9a8 + 42a’b + 49a°b?. 


MISCELLANEOUS EXERCISES 
Factor if possible : 


= 


oy — 2. 

—ab?c'd! + 3a4b?c2d + 6a°b’c?d?. Find two factors only. 
ax — bx + ay — by. . 
4a? — 12a + 9. 

a’b? + 4ab + 4. 

a’b? + 2abx + 2. 

an — 1. 

ab? + 10abry + 2527y?. 

x? + xy + Qu 4+ 2y. 

vt — 627 + 9. 

3a(a ++b+c) —(a+b+4+0). 

x§ — 27° + 1. 


areata tere See 


Bee 
war SS 


116 FACTORING [Cuap. XIII. | 


| 


' 


In each of the following replace the parentheses by a term | 


which will make the trinomial a perfect square: 


13. vy? +()+ 2. 14. 4a? + 4a+( ). 

15. ( ) + 42ab + 9. 

16. Find three trinomials which are perfect squares and 
which have 22?y for a middle term. 

17. Factor a? + 2ax’y + x+y’. 

18. Factor x?" — y’. 

19. Factor 2ax + 2ay — 3bx — 3by. 

20. Factor 169x* — 182a?y + 49y?. 


79. Trinomials of the form a? + (a+ b)x + ab. We have 
found that the product of two binomials having a common 
term is given by the formula 


(c+ a)(a+b) = 224+ (a+ b)x + ab. 


Here it is to be noted that the coefficient of x is the algebraic sum of 
a and b and that the last term is their product. For example, 


(2 +4)(¢4 +3) = 22 + 7x + 12, 
(x —4)(2 —3) =2? — 7x + 12, 
(cx —4)(2 +3) =2 —-z- 12, 
(c¢ +4)(@ —3) =2? +2 —- 12. 


If a trinomial comes under this type, it is possible to find two 


numbers whose sum is the coefficient of « and whose product 
is the last term. These trinomials can usually be factored 
- by inspection. 

EXERCISES 


From the following, pick out those trinomials which are of | 


the form 2? + (a + b)x + ab and find the factors. 
BF ersten pit i), 


Sotution: We are to find two numbers whose sum is —5 and whose 
product is 6. Such numbers are —2 and -3. Hence, a? —5¢+6= © 


(x — 2)(a — 3). 


{ 
fe 
| 


“Arr. 79] EXERCISES 117. 


2; 2? + 52 + 6. 
Bee a G, 


: Sotution: We are to find two numbers whose sum is —1 and 
whose product is —6. Such numbers are 2 and —3. Hence, x? — x —6 = 
(x + 2)(x — 3). 
4, 2? +2 -6. 
5. a + 8x + 2. 
SoLuTIon: It is impossible to find two integers whose product is 2 and 
whose sum is 8. 
6. 227+ 7x + 2. 9. 2? — 2x — 2. 12. a* + da-+ 1; 
7. 224+ 32 4+ 2. 10. +2 —-2. 133 a ae: 
8. 2? —3r44+2. 11. a@+4a+1. 14. y?> + 6y + 8. 


MISCELLANEOUS EXERCISES 


: Factor : 
im 1. 4x(a — b) — 3(b — a) + 5y(a — b). 
2. 22 — 62+ 8. 6. ax? — 8ax + 1da. 
a. zc — 22 — 8. 7 n+n— 12. 
4, 6? + 2b —- 8. Sip at 14": 
5. bry + 4by — cry — 4cy. 9. p?+7p+3 


10. 2? +47 +3. 


11. Fill out the parentheses to make ( )+4ay+y? a 
perfect square. 

12. Fill out the parentheses to make 8la’x? + ( ) + 4a‘ a 
perfect square. 


Factor : 
3. 5 + 62+ 27. 16. az? + Yar + 20a. 
14, a? + 122. — 28. 17. 9a? — 54. 


15. 274 — 2° + 40 — 2. 18. y? — 4y — 21. 


118 FACTORING [Cuap. XIII. | 


80. General quadratic trinomial, ax? +ba+e. If we mul- 
tiply together the two binomials 3a + 5 and 2a + 3, we find: 


3x +5 3D eo 

24 +3 x 

6x? + 10x 2x 3 
9x + 15 


622 + 19x + 15 


The two products 27-5 = 10% and 3xz-3 = 9z are called 
cross products. 
The product 6x? + 19x + 15 is a trinomial of the form 


O07 be: 


While the product of two binomials like 32 +5 and 2% +3 
is a trinomial of this form, yet all trinomials of the type 
ax? + bx +c cannot be factored into such binomial factors. 
If the trinomial can be factored it is often easily done by 
inspection. 

Example. Factor 67? + 192 + 15. 

The product of the first terms of the binomial factors is 6x2. The 
first terms are then 2z and 32, or 6x and 2, if the coefficients are integers. 


The product of the second terms of the binomial factors is 15. The 
second terms are then +3 and +5 or +1 and +15, if the second terms are 


integers. Since the middle term of the trinomial is positive we keep only 


the positive terms 3 and 5, and 1 and 15. 

We have now to pick out two binomials having 2x and 32, or 6x and 
x for first terms and with 3 and 5 or 15 and 1 for second terms in such a 
way that the middle term of the product is 19z. That is, the binomials 


are chosen so that the algebraic sum of the cross products is 19%. By | 


trial, we find the factors to be 32 + 5 and 2z + 3. 


EXERCISES 
Find the following products : 
127 +3) Tote 4) 5. (2a + 1) (8a + 1). 
2. (22 — 3) (52+ 4). 6. (2c — 4) (2c + 8). 
3. (2% — 3) (5% — 4). 7. Qe +5) (22a 
4. (2x + 8) (5a — 4). 8. (Sy + 4) (2y + 8). 


113 


6x? + 7x + 2. 
6a? — 8x + 2. 
62? — 7x + 2. 
10x? + lla + 3. 
10a? — 132 + 3. 
1527 4 132 Ps: 


Show that the trinomial x? + (a + b)x + ab is a special 


z*.— (2 — 18: 
zy? — 15a — 34. 


PP — 27 — 7 + I. 


4. 8x7? + 227 + 15. 


8a? — 23x + 15. 
?+(x+y)t + ry. 


Arts. 80, 81] EXERCISES 
Factor the following : 
9. 277 + 7x 4+ 3. 15. 
10. 322 + 5x + 2. 16. 
11. 527? + 7x + 2. 17. 
12. 5a? — 7x + 2. 18. 
13. 227 — 5¢ + 3. 19. 
14. 2x? — 7x + 5. 20. 
® 21. 
ease under the type az? + bx + ¢. 
Factor: | 
22. 22+ 92 + 14. 24. 
) 23. 2? — 4x — 21. 25. 
| MISCELLANEOUS EXERCISES 
1. 2(xe+y)+a(x+y). 3: 
2. 9(x + y)? — 4. 
5. 8x? + 23x + 15. 
6. 4x(a + b) — 8(a + b) + Sy(a +). 
7. p?>+4p — 21. 14. 
8. a? + 9ab + 80. 15. 
9. a®b? — a®b? — ab+ 1. 16. 


cation, 


and 


ax + by — ay — bx. 


- (a+b)? — ee. 
. (24 + 3y)? — 1. 
. 8a? + 1212 4+ 15. 


81. 


iy & 
18. 
19. 
20. 


3x? — bax? + x — 2a. 

re tgs WPA Bae a Ua 

82? + 43x + 15. 
(a+b+c)?— a. 

4ax® + 8ax — 8a — 4az’? 


Sum and difference of twocubes. By actual multipli- 


we find 


a+b) = (a+ b)(a@ — ab + db’), 
a — B3 = (a — b)(a@ + ab + Db’), 


Any expression which may be written as the sum or difference 
of two cubes can be considered as the product of a binomial 


factor and a trinomial factor. 


120 FACTORING [Cuap. XIII. 


EXERCISES 
Factor the following : 
1. x? + 27. 


SoLution: We may write the expression in the form z? + 33 from 
which we get x? + 33 = (x + 38) (2? — 32 4+ 32), 
v427 = (x4 + 3)(2? — 32 49). 


The factor 2? — 3x +9 cannot be factored since there are no two integers 
whose product is 9 and whose sum is —3. 


2, v= 24. 6. 8 —1. 

3, a — 1. 7. Sa° = ize: 

4. a +1. 8. y® — 1252%. 

5. § +1. 

9. a’ + b°. Find two factors only. Hunt: b%= (b?)3, 


10. x? — y®. Find two factors only. 
11. 2? — y’. Find two factors only. 
L2e or 
Hint: This expression can be written as the difference of two cubes 


or the difference of two squares. Factor by both methods. Which is_ 
the easier? 


13. 64275 — y, 

14. a® + 6°. Find two factors only. 
15. 1252%y? + 82%. 

16. x’ + y". Find two factors only. 
17. a” — x§. Find two factors only. 
18. x" — 1. Find two factors only. 
19. x" + y’". Find two factors only. 
20. (a + b)? + (a — b)3. 

21. (a — x)* — 2. 

22. (a — x)? + 23. 

23. (x? — 1)? + (a? + 1)%. Find four factors. 
24. 125(a + y)? + 82%. 

25. 1000 — 1. 


Anrs. 81,82] | SUMMARY OF FACTORING 121 


26. Give in words a rule for factoring the sum of two cubes. 
27. Give a rule for factoring the difference of two cubes. 


82. Summary of factoring. No simple general rules for 
factoring can be given, but a few suggestions will be helpful. 

(1) First take out all monomial factors, not forgetting factors 
expressed in Arabic numerals. 

(2) After the monomial factors, if any, have been re- 
moved, the number of terms will usually be the best guide in 
factoring further. 


(a) Binomials are factored as — 
The difference of two squares, a? — b?. 
The difference of two cubes, a? — 6°. 
The sum of two cubes, a’ + 6°. 


(b) Trinomials are factored — 
As trinomial squares, a? + 2ab + b?. 
By inspection, x? + (a + b)x + ab, and ax? + bx +. 
(c) Polynomials of four or more terms are usually factored — 
By grouping. 
As the difference of two squares. 

(3) Sometimes an expression needs to be rewritten in order 
to show the type of factoring. Before concluding that an ex- 
pression cannot be factored, see if an arrangement of terms will 
bring it under any known type forms. 

(4) Test each factor to see if it can be factored furthoe 

(5) It is convenient to remember that x? + y’, x? — ry + y’, 
w+ ay + y?, 24+ cy — yy’, 2 -— zy — y’ are prime. 


MISCELLANEOUS EXERCISES 


Factor: 

: 1. ax + ay + 2abz. 5. xv? — 216y°. 
2. 2a? — 2m’. 6. 348 4+ 1. 

3. 2? + y? — Qay. Tape: br. S. 
4, x? + 11x — 42. 8. 27a%b? — 18ab?. 


122 


21. 


FACTORING 

(4 — 92) P(e 15. 

Nets es ted be “16. 
. a +192 + 18. 17 
. 9-6a+a’. 18 
. 62? — 18x — 28. 19 
tei 0. 20 


Sat ot 


[Cuap. XIII. 


dx — Sry? — Bry + 10zry?. 
a® + 2a* + 1. 


. x — 8x'. 

. 16 + 8ab + a’b?. 
. 62? + 3x — 3. 

. 12 — 10a — 2a’. 


Hint: Collect the terms thus, (1 — x*) + (82? — 3x) and factor expres- 
sions in parentheses. 


22. 
23. 
24. 
28. 
29. 
30. 
31. 
32. 
35. 
36. 
37. 
38. 
39. 
40. 
41. 
42. 
43. 
51. 
52. 
53. 
54. 


a’ + 2a? + 4a + 8. 
x? — 132? + 36. 
e—x+ uy — y. 
a®b®ct — 125c?. 


26. 
27. 


25. 4a? — 20ab + 46. 


26x? — 63x — 5. 
4x? — 28xry + 49y?. 


Find four factors. 


lla? + 9x2 — 2. 


xty — xry'*. 

24a4y? + 26 + 14427y*. Find four factors. 
5abx® — 5a?x?. oo: 

16x + 8abe + ax. 34. 


Sq} =. q'2, 

622 — x(a + 2) — (a 4 2)?. 
15a? — 142° — 82°. 
v—a+u-—a. 

25 — (2? + 2ry + y?). 

36 — 24 + 2a?y? — y®. 

16ab — 24abx + Yabz?. 
343p3g? — 729p°. 

5la? — 45a — 6. 

50x? + 60xy + 18y?. 

(24 +3) (w@+y) +4(a +9). 
4a3 — 3ry — 8a?y + By’. 
Aa’b? + 36a7ba + 8laz’. 


55. 
56. 


3a° + 375a?. 


. 2xry® — 102?y? — 282°. 

. a+ (b— 2b2?)ay — 2b?2y8 
. 5a* — 5a*b — 5ab — 5a. 

. L— ab? — xy? + 2absy. 

. «2 — (a — b)x — ab. 

. 22a? + 59x + 39. 

. (Qa +3)3 — (Qe — 3). a 


(32 + y)® — (2% — y)*. 
7 — 6% —Z4ee 


“Arr. 82] EXERCISES 123 


57. 
58. 
59. 
60. 
61. 
62. 
63. 
64. 
65. 
66. 
67. 
68. 
69. 
70. 
ais 
12. 
73. 
74. 
75. 
76. 
77. 
78. 
79. 
80. 
81. 
82. 
= 83. 


7x? — 21cx — 280c’. 

9a? — 6a? + a’. 

(a? + 2ab + 0?) — (a? — 2Qay + y?). 
xv? — Qay + y? — 49. 
xy—-l+a-y. 

sr + st —? — rt. 

320° — 48x?y + 182y”. 

2 — a? — b? + y? — 2ry — 2ab. 
e+24e. 

n? + 4mn + 4m? — 16. 

2ry —2?—-y +1. 

Qy? — a(x + y). 

8 + (x — 2). 

x§ — 1324 — 902’. 

2abx? + 10a*bx — 28a%*b. 

Sa2x? — 56a?xy + 98a7y’. 

16924 — 1562%y? + 36az%y°. 

27a*" — 125. Find two factors only. 
a> — ab? — a’b? + 6°. 

3 — 27 — 9a? + 272. 

4+ y> + 4a? 4+ dry — y’. 
ep+y+ta— yy. 

2524 + 102° + 2”. 

28abn? + 384abnm — 12abm?. 

Sy’? — 80y?x + 200y”. 

3 — 2 +323 — 2°. 

6x? + ( ) — 14. Find three different expressions replac- 


ing the parentheses which will permit the trinomial to be 
factored. : 


84. Write down at random three trinomials of the form 


ax? + bx + c and try to factor them. 


124 


FACTORING [Cuap. XIII. | 


Fill the following parentheses so that the trinomials will 
be perfect squares : 


85. 49a2x? + 1l4ax + (\). 88. ( ) + Garry + y’. 
86. ( ) + 12pqrx + 2’. 89. 22+ Qryz+(). 

87. 1607+ ( ) + 49y’. 

Factor : 

90. 5a? + 10x — 5a” — 10. 93. (1 + a)? — (1 + 2a)’. 

91. 2? + 38y — 3x — zy. 94, (x + 2)? — 25(x + 3). 


92. 144v?y?2? + 24ryz 4+ 1. 
95. Write down three perfect square trinomials whose 
middle term is 4abe. 


Factor : 

96. 100ax? + 90a?x — 90a’. 

97. 10x? — 2524 — 2”. 

98. ax? + ba? + aty? — bty?. 

99. — 2cy — 3bz + 4ay + cz + Oby — 22. 
100. atx? — b’x? + aty? — b*y’ 


101. 
102. 
103. 
104. 
105. 2° 
106. 2” 
107. 
108. 
109. 


eo+y+aeyt+aoay+rt+y. 

B4ty+te—syt+y’. 

a® + 2ab + b? + 3a + 3b. 

a‘(a + x)* — (ax)*. 

—~ay —xvyt+y’. 

— 2(a — b)x — 4ab. 

ax? + ba? + ax + bx. 

Show that x? — 16 is a special case under az? + bx +. 
What values have a, b, and c when 2? — 32 is consid- 


ered as a special case of ax* + bx +c? 


Factor : 


110. 2ry + 3yz2 + by + xz. 


Bik 


(e+ y +2)? —1. 


112. 49p%q? + 42pq + ? = a perfect square? 


EXERCISES 125 
. y — 8y — 65. 117. (a + b)§ — (w@ + y)3, 
toe or — 12. 118. 82? + 297 + 15. 
meee re ts 2. 119. 8a? -- 62a 15: 
- 1728c3 + 643. 120. 82? — 29% 415, 


- (a+b)? —-(@+y +2)? 

- (2+4+2)? -(24+274y)2. 

. 4abed — 24cdx — 3abmn + 18mnz. 

. ov +1 — 22 — Be. 

- 2ax? + 8ary — 2bary — 3by?. 

-9-(8-—2-y)?. 128. 82? + 267) 4-15: 
. 8x7 + 34x + 15. C29 a | 

: (a + 6)§ — (a — 6), 

131. 


In arithmetic we find that the sum of the odd powers 


of two numbers is divisible by the sum of the numbers. What 
case of factoring comes under this rule? 


132. 


What case of factoring comes under the rule that the 


difference of the odd powers of two numbers is divisible by the 
difference of the numbers? 


Factor : 
133. a + bx + ay? + bay’. 
134. 10% + 10y — 5ax — 5ay — 15a2x — 15a?y. 
135. x** — 1. Find two factors only. 
136. 3a* — bab — 4a — 8b. 
137. a° +1. Find two factors only. 
138. 20m? — 9nm? — 20n2m?. 


139. 
140. 
141. 
142. 


ae “Ere  F..4e 


x? + 2a*x + 384 + 6a?. 

492? — (1 + 2? 4+ 2)? 

a? y? 4+ ax? +4 ay? + aa? + ay? 

ax + ay + az + bx + by + bz + cx + cy + cz, 


CHAPTER XIV 


EQUATIONS SOLVED BY FACTORING 


83. Quadratic equations. If, in 2? — 4x + 3, we substitute 
different numbers for x, we find different values for the expres- 
sion. Thus, when x = 0, the value of 2? — 4a +3 is 3; when 
z = 1, its value is 0; when z = 3, its value is ?. We give a table 
showing the value of the expression for all integral values of a 
from —4 to +4. 

Suppose we ask what values of « make 2? — 4x + 3 equal to 
zero. By referring to the table 
we see that the expression is 
zero for at least two values of 
a; that is, when x is 1 and when 
az is 3. We have really asked 
a question equivalent to the 
following: What values of 2 
satisfy the equation 


w—4r+3 =0? 





This equation is different from 
the’ equations previously con- 
sidered. In the left-hand side we find one term containing 2, 
one containing z, and a term containing no unknown. In 
other words the equation a? — 4x + 3 = 0 is a special case of # 
class of equations of the form az? + bx + ¢ = 0, where a =], 
b = —4, and c = 3. 

Equations which can be reduced to the form az? + bx +¢ = 0, 
in which a, b, and ¢ are any numbers whatever, except that a 


cannot etn zero, are called quadratic equations. 
126 


PO > fy 


Arts. 83, 84] | EXERCISES 127 
EXERCISES 


_ Make tables showing the values of the following expressions 
for the integral values of x from —4 to +4. Indicate if possi- 
ble those values of x which make the expressions equal to zero. 


1. x? — 34+ 2. 6. 277 + 8x — 10. 

oe — 2. ,. Y ev pies Si geod © 

aa r — 6. Fee rare AB 

a. x? — 7x + 12. 9. (v + 2)(2x — 8). 
B..2? + x — 12. L005 ( = 6) tz — 3): 


_ Reduce the following quadratic equations to the form - 
w+ br+c = 0: 


a1. x? — 3x = 2: 16. 27 — 52? = 32 — 722. 
12. 277 —- 444+ 7 = 9. 17. 13+ — 4a? = 27? +4741. 
is. 2+27 = 2, 18. 2 — 77 + 2? = 27? + 2. 
a4. OF = (2x? — 4. 19. 2 -—3r+4=4 —- 3z. 
16. 77 + 3827-1 =2. 20. pv? +qzut+r=retqt+re. 


«684. Factoring applied to the solution of quadratic equations. 
Jo solve a quadratic equation, it is not necessary to proceed 
y trials as in Art. 83. If the quadratic expression can be 
asily factored, the method illustrated by the following example » 
an be used. 





_Example. Solve 2? —- 4% +3 =0. (1) 
SoLuTioN: Factoring the left-hand side of this equation we find 
(x —3)(x — 1) =0. (2) 
The products of these two factors is zero if 
- 2-3 =0, orifz —1 =0. (3) 
From (3), . #& =3or 1. 


The equation has then two roots, 1 and 3. Checking by substitution, 
(1)? -4-14+3=0,. 
id (3)? -4°3 +3 =0. 


= 


128 EQUATIONS SOLVED BY FACTORING [Cuap. XIV. 


If in any product, any factor is zero, the whole product is 
zero. Conversely, if any product is equal to zero, some factor 
of that product must be zero, and any factor which contains 
an unknown may be equal to zero. Therefore, in solving any 
quadratic equation in which one member is zero and the other’ 
member can be factored, we find values of « which make each 
of the factors zero. That is, we may equate each factor to zero) 
and solve for the unknown. 





Thus, to solve 2a? = 7x —5, we write the equation in the form 
or — 72 +5 =0. Factoring, we have (2% — 5)(x —1) =0. Equating 
each factor to zero, 2c —5 =0, x —1 =0, and we find x = 5 and x =f. 


EXERCISES 
Find two solutions for each of the following quadratic 
equations : | : 
1. (@@ —2)(¢— 7) = 90. 10. 22? + 22 = lige oe 


2. (x +3)(a —1) = 0. 11. 42 +2? = Alger 
3. (2x + 5)(a + 6) = O. 12. 6 — (i374 

4. (5a +6)(22 —4) =0. 18. 32? — 14% = 2° + 60. 
5. (x +1)(82 +7) = 0. 14° 2° =a 

6. 2? —324+2 = 0. 15. 2? 40; 

7, a + 7a — 30 = 0. 16. 42? = 81. 

8. 2? + 132 = 30. 17. 22? + 5¢ =3 = 
9. 


g? 4102 +24 =—-—427. 18. 6 = 62 + 352. 


Solve first for x and then for m in Exercises 19 and 20. 
19. 22? = 3mx — m’. 
20. x2 — 9max + 14m? = 0. 
21. a? — Zab + 10b? = 0. Solve for a and for b. 
@99. 77? + 13% = 8 + 32. 
23. 202? + lla —3 =0. 
24. 2a +a —3 =0. 








Arr. 84] PROBLEMS 129 


25. 2? + mx+3x+3m=0. Solve for zx. 
\) 26. 2? — mz —nzr=0. Solve for x. 
‘) 27. 2? — 4a? =0. Solve for x and then for a. 
| 28. 2? +ax+bx+ab=0. Solve for x. ~ 
im 29. x? + px =0. Solve for x. 
, 30. av?+axr+br+b6=0. Solve forx. 
31. ax?+abr+2+b6=0. Solve for z. 
ly 32. az? + acr + bx + bc =0. Solve forc. 


PROBLEMS 

The following problems involve the solution of quadratic 
quations; find all solutions possible for 
‘he equations and determine whether or 

‘ot each solution is a reasonable result: y 





b 
1. Inaright-angled triangle the longer 
+g is two feet shorter than the hypotenuse = 
ut two feet longer than the shorter leg. c= a7 40° 
Vhat is the length of the longer leg? Fra. 20 
SoLuTion: Let x = the length of the longer leg. 
hen x — 2 = length of the shorter leg, 
ad x + 2 = length of the hypotenuse. 
rom page 80 ( +2)? = 2? + (x — 2)2, : 
(See Fig. 20, also Fig. 15). 
ence m@+4e4+4 =a 422? —-47 44, 
x’ — 8x = 0. 
hence 2 =, or  =-8; 


The answer x = 0 must be cast aside, for it has no interpretation when 
iplied to the problem in question. 


2. In a right-angled triangle, the short leg is two feet 
orter than the hypotenuse and one foot shorter than the 
oger leg. ‘Find the length of the short leg. * 


3. A positive number when multiplied by a number 5 
nes as large becomes 405. What is the number? 


130 EQUATIONS SOLVED BY FACTORING [Cuap. XIV. 


4. The square of a certain number plus twice the number 
itself is equal to eight times the number. Find the number. 

6. The product of two consecutive integers is 306. What 
are the numbers? 

6. The sum of the squares of two consecutive integers 
is 41. What are the numbers? 

7. The sum of the squares of three consecutive integers 
is 50. What is the smallest of the numbers? 

8. The square of a number is 20 more than the number 
itself. What is the number? Is there more than one answer? 

9. The area of a rectangle.is 18 square feet. The length 
is one foot longer than 4 times the width. What are the di- 
mensions of the rectangle? 

10. A rectangular floor is 4 feet longer than it is wide, and 
its area is 320 square feet. What are its dimensions? 

11. The perimeter of a rectangular field is 60 rods, and its 
area is 200 square rods. What are its length and width? 

12. Make up a rectangle problem whose solution will in- 
volve a quadratic equation. - | 

13. A photograph is one inch longer than it is wide. It 
is mounted on a card so that there is a 1-inch margin on all 
sides. The total area of the margin is 2 square inches greater 
than the area of the photograph. What are the dimensions 
of the photograph? 

14. It takes 96 square inches of paper to cover a cube. 
What is the length of one edge of the cube? 

15. The dimensions of a closed rectangular box are con- 
secutive integers. The entire outside surface of the box is 
52 square inches. What are the dimensions of the box? 

16. A paving brick is 4 inches longer than it is wide. The 
thickness is 4 inches. The volume of the brick is 128 cubic 
inches. What are the length and width of the brick? 

17. A rectangular box is 5 times as wide as it is deep and 
twice as long as it is wide. The total surface of the box is 18C 
square inches. What are the dimensions? | 

| 
| 


‘Arr. 84] PROBLEMS 131 


18. A rectangular solid is twice as long as it is wide, and 
the width is 3 inches more than the depth. The total surface 
is 160 square inches. Find the dimensions of the solid. 

19. Make up a problem about a rectangular solid or brick 
whose solution will involve a quadratic equation. 

20. A club had.a dinner that cost $60. If there had been 
4 persons more, the share of each would have been 50 cents 
less. How many persons were there in the club? 


CHAPTER XV 


HIGHEST COMMON FACTOR AND LOWEST COMMON 
MULTIPLE | 


85. Greatest common divisor in arithmetic. The largest 
integer contained as a factor in two or more integers is called | 
in arithmetic the greatest common divisor. This number is 
easily found by separating the numbers into their prime factors 
and multiplying together those found in each number. ‘Thus, in) 


12 =2°2°3, 18 =2°3:°3, 3d0= 2a 


the greatest common divisor is 2:3 = 6. 















86. Highest common factor. In algebra, a number or 
expression which is a factor of each of two or more expressions, 
is called a common factor. Thus, in 102?y, 4xy, 8xy?, the com- 
mon factors are +2, +27, +y, =ay, ==Z2ay pus the 
positive factors alone are considered. The factors 2, x and v 
are the common prime factors. ; 

The product of all the common prime factors of two or. 
more expressions is called their highest common factor. 
(H.C.F.). For the above expressions 102*y, 4xy, 82y’, the 
H.C.F. is 2zy. Ww 

Two expressions that have no common factor except 1, are” 
said to be prime to each other. 

In algebra, the word “highest” is preferred to the wort ' 

‘“‘oreatest’’ in connection with common factors. The highes' i" 
common factor is the common factor which contains the highe 
number of prime factors. In the above example, 2zy is the 
highest common factor, but if « =1 and y = 3 it is not th g 
greatest common factor. 

132 


Arr. 86] EXERCISES 133 


EXERCISES 


Find the highest common factor of the following sets of 
expressions : 
1. 8a‘b3c?, 4a7b?c?, 16a2bic!, 
The common prime factors are 2, 2, a, a, b, b, c, c. Hence 
the H.C.F. is 4a2b2c?. 


UL ye, Ory" 2°. 5. dmn*p*q*, 10m n’, 2Omnpq. 
pada 210*b’c*. 6. 12a°b?, 6a7b®, 18a%b?, 9a‘b4. 
4. 3a’b, bab, 3b’. G Say, 16a", 92747 7": 

8. 14abe, 8a*b?, 7ab?c?, 2a5b?. 

Pot yor, y*, 1 2r5y°- 249%,)4, 


10. 2a? — 2ab?, 4a? + 8a2b + 4ab?. 


SOLUTION: 2a’ — 2ab? = 2a(a? — b?) = 2a(a +b) (a — b). 
4a5 + 8a*b + 4ab? = 2:2-a(a +b)(a +b). 
The common prime factors are 2, a, (a +b). 
Hence, the H. C. F. = 2a(a + b) or 2a? + 2ab. 


11. 2? — y’, x — 2ry + y?. 
iz. 2 — y*, x? — 7’. 
13. 2+ y', vy + ry’. 
14. 2 — y*, w+ xy + ry’. 
15. a? + b?, 2a? + 4ab + 2b?, a? — b?. 
16. av —y + xy — a, ax? + ay —-a—y. 
17. vy —y — 22+ 2, ry —2z2—y —z@. 
18. 1584, 1728. 
19. 861, 615, 984. 
20. 156, 130, 182. 
21. 9 — 2x’, x? — x — 6. 
ieee. 0 — 27, 7? — x — 2, 2x? — x — 6. 
im 23. 2* — xy", 32° — xy — Qry?, x? — Qa%y + ay? 
24. ax + a’y + abx + aby, 2ax? — 2ay?, 3ax* +6axy + 3ay?. 
25. abx? — abxry + aby’, abx? + aby®, aba? + abry + b?y?. 


134 H.C. F. AND L. C. M. [Cuar. XV. 


87. Lowest common multiple. An expression which con- 
tains each of two or more given expressions as a factor is called 
a common multiple of the expressions. Thus, if xz, xy, 102, 
2y, dxy” are given, then 10zy? is one common multiple. There 
are many more, for example, 20x?y’, 100zy*, 402%y%z?. | 

Among all the common multiples of a given set of ae | 
sions, there is one which is most important. It is called the 
lowest common multiple (L.C.M.). The lowest common mul- 
tiple of two or more expressions is the product of all their’ 
different prime factors, each factor being used the greatest 
number of times it occurs in any of the expressions. For 
example, consider the expressions x, xy, 10x, 2y, 5ay?. In 
por the different prime factors are xz, y, 2, 5. The factors 2, 

2, and 5 occur only once in any expression. The factor y occurs . 
hates in 5xy”, since we may write it Sry: The L.C.M. is then 
2-5-xa-y-y = 10zy’. ) | 






EXERCISES i 
1. What is the least common multiple of two or nal 
numbers in arithmetic? 
2. In connection with common multiples in algebra wh y 
should the word “lowest” be preferred to the word “ least’? 
Find the L.C.M. of the following sets of expressions : : 
So LY Bae aes 8. ay? a eee j 
4. 3xryz, 5ax, 10a%2°. 9 («+y),24+y*, 2 — y’. q 
6. 12a%bc*, 6ab*z*, 8abc’. 10. x? — 77, a 4 Qay ey, 
6. (n’m'*, 13an’m, 2amn. 11. 2° 4+ 32 +2) ae ee | 
(Peni Gn lponnl an 12. 6x? + 13a + 6, 42? — 9. 
13. 32? — 10% 4+ 8, a? — 44 4 4. 
14. ab — Bb, a + ab, ab + DB. 
15. 67? — 5% + 1, 8a? — 62 + 1, 102? — 7x +1. 
16. anes 3x? + 8x + 4, oo ae 
17. — y*, 2 — y*. 
18. uk ee 


Arr. 87] EXERCISES 135 


19. 244+ zy, — xy + ay,2+ y. 
20. a — b, (a? — b?)?, (a + Bb). 
21. ax? + ax, ax* + 3ax 4+ 2a, 2? + 4¢ 4+ 4. 
22. 3(ab? — b°)?, 2b(ab — x?)3, 6(a2b? — a). 
23: x2? — xy — Sax + By, x? — 10x + 25, 2? — 25. 
24. av + a+ 2x +2, 5a? + 7a — 6, 5ax + 5a — 3x — 3. 
25. Gar + 9a + 4x + 6, 2ax — 4x 4+ 3a — 6, 3a? — 4a — 4. 
26. a*b — ab*, 7a? + Tab + 7b?, 14a3b + 14a2b? + 14ab3. 
27. 1—-m+n—mn,1—m—2x4+mz7z,1+n—2— ne. 
28.2? +2? 4+274+1,2724+ 7, 7? + 27 +1. 
| 29. ax+y — by + ay — br +2, ax + ay — bx — by, 
¥—ab+a-ab+0b? -b. 
30. ax +a — bx + ay — b — by, (x+yP+ut+y, 
w + ay — by — bx. 


a a® + 63, ac — be + ad — bd, a’c + abc + b’c + a’d + abd 
| ae 


CHAPTER XVI 


FRACTIONS 


88. Fractions in arithmetic. If a unit be divided into 7 
equal parts and 3 of these are taken, we denote the amount 
taken by #. This fraction may also be understood to mean 
3+ 7. 

The fraction # is thus the answer to the questions : 

1. What part of 7 is 3? 
2. What is 3 + 7? 
But a fraction such as 5 cannot be regarded in the first of the 
4 F 
two ways mentioned, since a unit cannot be divided into 22. 








equal parts. The fraction a is thus taken to mean. 3 + 2% 


that is, the fraction x is an indicated division. 
4 


89. Fractions in algebra. Similarly, in algebra a fraction 
is an indicated division in which the dividend and divisor are 
algebraic expressions. Thus, 4 
60-3) C(t 
iT bc t4on ee 





are fractions. 


The dividend is called the numerator, and the divisor the 
denominator of the fraction. The two are often called the 


terms of the fraction. The fraction i is read “x over y,” 


“x divided by y.” 
The rules for fractions used in arithmetic apply 1 in alechil 


In particular, much use is made of the 
136 


i. 


_ Arr. 89] | EXERCISES 137° 


. Principle. The numerator and the denominator of a fraction 
may be multiplied or divided by the same number without changing 
the value of the fraction. 





Zita 4 
a, 373375 
a, 141014 7D 
Similarly, a oa7 73 
: 
i y pkey © yesh gre a 
and e-8 (@-b)+(a-b) a+b 





(a-b? (@ —b)? +(a—b) ~a_-b 





2 EXERCISES 
1. State two meanings of : 7 0G - of a and of © : g 


‘if a, b, and c denote integers. 

2. In his four-year course, a student spends for books a 
dollars the first year, b dollars the second, ¢ dollars the third 
and d dollars the fourth. What fraction denotes the average 
expense per year? 

3. What part of a piece of work can a man do in one day, 
if he can do the entire work in 6 days? In 53 days? In 
a days? 

4. The grades of a student in two subjects differ by 5 
points. The lowest grade is x. What is the average grade? 

By multiplying or dividing both numerator and denom- 
inator by certain numbers, replace the following fractions by 
equal fractions. Give at least three solutions for each exercise, 





| 7 2 r+y —14 
| | 5. 3" 8. 5 iB Ee zy 14. ee 
| 4 x a ab 

| 6 9 9. y 12. a BS b 15. ab 
‘ m2 2 

| ae TAS tre ies Eee fonts 


2° " BF * ab? cd? 





138 FRACTIONS “=| Crapo x V ies | 


90. Division by zero. The numerator of a fraction may 
be any number whatever, including zero. In the case of the 
denominator, however, we have one exception. The denom- 
inator cannot be zero, for division by zero is excluded. That is, 
an expression. with denominator zero is not considered as a 
number. Care must be taken not to give such values to the 
x+4. 
73s 
a number for any value of x except x = 3. For x = 3, we have 


letters as will make the denominator zero. ‘Thus, 


5 which has no meaning. 


EXERCISES — 


Give the values of x for which the following expressions 
have no meaning : 

















+2. 3a + 4: 1 
a pe e 32 — 4 4 a(x — 1) 
x 1 3°. 52a a} 
4 c+1 a; rt yi 3x + 5a? 
x+1 x+4 2 
Se ct : 3 f (2x — 1)(x + 2) 


91. Signs in fractions. In any fraction, there are three | 
signs to consider: the sign before the fraction, the sign of the 
numerator, and the sign of the denominator. Thus, 












i 7. 

Historical note on fractions. . Fractions offered very great difficulty 
to the ancient nations. In their operations with fractions, the Babylo- 
nians reduced all fractions to the denominator 60. Similarly, the Romans 
reduced fractions to the denominator 12. The Egyptians and the Greeks 
reduced fractions to the same numerator. In fact, Ahmes (see p. 80) 
seems to have restricted the term fraction to those with numerator equill 
to 1. Other fractions such as 3 were expressed as the sum of fractions 
with.numerators equal to 1. Thus, “‘} and }”’ took the place of 3. This 
probably seems rather awkward to us, but it shows that eadinngs offered 
real difficulties to the early students of mathematics. 





| Arr. 91] ' SIGNS IN FRACTIONS 13973 


‘have the three signs attached. 

Since the names numerator, denominator, and fraction mean 
dividend, divisor, and quotient respectively, the laws of signs 
for wes must hold for fractions. , Hence, we have 


2 SCRE 
b pies hin = Ach eh 


See V2) ee (2a) ow 
; -($) = +55 - (5) = +2; and eae b 


In words, we have: 


(1) Any two of the three signs affecting a fraction may be 
changed without changing the value of the fraction. 


(2) The sign of a fraction is changed by changing the sign of 
either numerator or denominator alone. 


| 


Thus, * and = are both positive ; = and * are both negative. 

In changing the sign of the numerator (or denominator) 
‘when it is a polynomial, be careful to change the sign of cay 
term of the polynomial including the first. 





Thus, changing the sign of the fraction 
ty 
3x +y 

by changing the sign of the numerator gives 
-o+y 
3x +y 








EXERCISES 


Reduce to equal fractions with positive signs in both numera- 
tor and denominator : 
4 ae -2 =a). 


aye 2. ris , 3. ae 


{ 
| 
| 


140 FRACTIONS [Cuar. XVI. ! 














—4c 6 y+9d 
hear ih ae fa ahi SO: 3 
—x%-—2 —m — 2 —a—b 
Shea Bipir te eg eae a eee in ee 
10. Show that —. =—~—- Give reasons for each step. 
a—b b-a : 


11. Show that 2 sy as 
l-y y—-1 


2a i 2a 
bbs a) - Lee 





12. Show that 


92. Reduction to lowest terms. A fraction is in its lowest 
terms when no factor except 1 is contained in both numerator 
and denominator. 


Thus, : and = octet are in their lowest terms, while 2 and fee 
+] 8 e+: 


not. Pee a epee to its lowest terms we use the principle of Art. | 
89 and divide both numerator and denominator by the factors common | i 
to both. For example, 
v2—ax x(x —-1) oat 
24+ (ce +1) eee 





are 














EXERCISES 
Reduce the following fractions to lowest terms : 
21 Pape nk co ls xv — @ 

28 e 203 + 222 cc 2(x — ay. 
2 4ab?ct 5 30 — 8 P 21 + 10x + 2? 

" 2abe " a+1 ; v9 | 
3 250 6 aan 9, ae (v +4) 

Lgsio " 327 + 62% + Ox — 16 . 

ax? + ary 


ve ax’ — axy? 





! 
i 
' Art. 93] CANCELLATION 141 


93. Cancellation. The process of dividing the numerator 
‘and denominator by the factors common to both is called 
“cancellation by division. It is merely an application of the 
“principle of Art. 89. The Baoeed ure is illustrated by the 
following examples. 








Example 1. Reduce aan ; to lowest terms. 
wx De 
4 ae 14a4b%c6 V AB a’bc? 
i SOLUTION 2 abc “3 Tepe 3 
Example 2. x3 -2y? 2x(e—y)(x + y) z+y 


a — xy) ey) (x2 +2y +y*) x(a? + ay +9) 
x 


- , ‘ 
‘Cancellation can be used only with factors of the numerator 




















‘and denominator. 
4 ae GS Y) 2+2 
_ Thus, in the fraction a (ray We may cancel the 2’s, but in 
| “(i +y) 2+ 
we cannot cancel the 2’s ; a here 2 is not a factor of the numerator 
or denominator. To illustrate, H = at To strike out the 2’s gives 
5 ig, 5 
3 But gis not equal to 6 
EXERCISES 
Reduce to lowest terms: 
4,5 4 
2 he Be 6. r 
xry 44 4% 
6 + 3x 
ae rogue 
ab 9+ 3x 
3, Tab%ctds Rian, 
"  25be?d3 tay te be 
4 28xry?2° 9 4+ 20 
"14226 ' 24+2 
14 — 7x 
Seca 10 


* Qe + Qry eT — Te 


142 s/ FRACTIONS [Cuar. XVI.4 














11, Bet Sy, 18, 2 
" 3x — by ' az + 3x + 2a+6 
2+ 14y oe 
12. 7 Dae 
4203 + Way a® — ab? 
i Bs 14y 20. a® — ab? 
14 Mateo a1, Aaah + Sate + Bot 2 Yh 
° 2 Quy + y’ , 2ax + 3a (2X4 dau 
8a2 + 8a + 2 ax + 3a — bz — a0 
15. 16a2a a a? .+-.bo 40 
16, mmm gg BOax + 45ay — 4a — 5y { 
* mn? — 2m?n + m? c 8x + 10y j 
ab — 36b 12ax — Gay — 50ca + 25ey 
Agha ee 24. ——— ee 
a? + 6a 227 + xy — y? | 





94. Reduction to common denominator. Two fractions 
are said to be equivalent when one is the result of multiplying 


or dividing both terms of the other by the same number. 


2 
Thus, : and < . an = = and — =? are pairs of equivalent fraction | 










Any set of fractions may be changed into a set of equivalent 
vegans all having a common denominator. 


5b ee 


2 5 Ge Soy Gy(a? +B) 
yO aed 


Thus, by’ Bby’ ‘Gby 





may be written 


This may be done in many different ways, but when th 
common denominator is the lowest common multiple of all the~ 
denominators the process is called reduction to the lowest com | 
mon denominator (L. C. D.).. The lowest common denominator 
of any set of fractions is the L. C. M. of the denominators. 


| Arts. 94, 95] EXERCISES 143 
EXERCISES 

Reduce the following sets of fractions to the lowest common 
‘ denominator: 


















































hes a | y 1 
Dey sy ye ) . 
25 8 * y-T yd) 
aay a’ 3a. SO a ne Le leer 
a+ba—b a. a+l1,a+2 
3. Zab? ) ab 10. 2, db b2 ) b3 . 
a+ba—b r+] 2-1 
| 2 aa 2 EG 
2x 3 a b 1 ab 
Pastimes 
: r+2 274+ 524+6 ee as a ab 2 
ee ag ca ga Bat 
aval ty y ty -—y ry-x 
eee 1 Ui, «2a ge Le aa 
x-1l2x+124-1 L-Y y-xuX ~-~u-y 
‘a? 3 
_@+5a4+6 2(a? +30 + 2) 
| ai rn 
ary a — yp CY x+y 
17. a +2, m + 4i. 
18. 54 +a, a+b — 2; ale 
3 ike os 


95. Addition and subtraction of fractions. In adding or 
ubtracting algebraic fractions we proceed as in arithmetic. 


lust as 3 feet + 4 feet. = 7 feet, 
3 4 7 
| eae ite 11 
| a a a 


144 FRACTIONS [Cuap. XVI. | 


EXERCISES 


Perform the following additions and subtractions: 














1 2 ~ page 
isco = ee 
pe : acts a 
0. . 05 ee 6 ay 2x 
Slee eres = 
22 127 2 1. 1 ee 
Tin De eS at be see 
St seaee 3. 
Geri § 1 re 8 
cee amaaey a 8 8 eee 
SOS aaa 0. (2 
m mM nN n 1 


To add expressions involving different units, as 3 feet and 
24 inches, we reduce them to the same unit. Thus, 


3 feet + 24 inches= 3 feet + 2 feet = 5 feet. 


Similarly, to add } and % they must be reduced to the same 


denominator. Thus, 
3:1 it 2am Sie Le 
479 12> eee 
In like manner, 
aC std Oe ad + be | 
btd ba bd mabe 


Similarly, for subtraction. Thus, | 





mae Bt) Aa Sey — 3a  4y? — (Bry — 32) | 
y-1° 4y 4yy—1) 4yy-1) © See i 
_ 4y? — 3ry + 32 qi 
~ ayy = | 
When this is put in the form of a rule, we have : | 


To add fractions, reduce the given fractions to a common de- 
nominator, add together the new numerators and place this sum 
over the common denominator. 


rs 95, 96] 


EXERCISES 145 


A similar rule holds for subtraction except that we subtract 
one numerator from the other instead of adding. 


















































EXERCISES 

Perform the following additions and subtractions : 
wes i. 5 -. 

| ica ea ae 

| 3. 24% ; 13. fas =o ; 

_ 4 rete s “Strato 

| 5 Stet. 15. Stat et iti. 

i P or+6  3x+a 16. at 2a 
14 21 elie a — | 
Batsty (tet tae 
¢ aes “ Ga = Ge 
ee aot eta 

(1. T+ et 20, 24 2-—*. 


96. Multiplication of fractions. As in arithmetic the pro- 
ect of any number of fractions is the fraction whose numerator 
the product of the numerators and whose denominator is the 
‘oduct of the denominators. 


146 FRACTIONS | [Cuar. XVI. 


; ‘ ; lee 6610 es 
Thus, in arithmetic pe tea) is 
In algebra, 
2 £+3. £2242 Qte+3) - 2+T) 4 


z 2£+1 2 +4e4+3 ceFHDeraaet+l ze+1) 


Expressions not in fractional form can always be considered 
as fractions with 1 for denominators. 





EXERCISES 


Perform the following multiplications and reduce the an- 
swers to the lowest terms: 


7 4 3 co r+y BY 




















= 29 52 
prema 2G 10, 2%. OYE WY. 
25 go tO by aa? ba 
—3 2 27 me ee” 2 2 
3. ie LOR t At. ye 
3ab 4ax 22\* 9 f24\ eae 
Are A ae —. |e ad 
2ry 9b = S e (3) 
5b? 8ab —a\? (ab)? 
oot 25 - & at 
a bed x\? (x+y) 
6. aoe ee aS 14. x (2) 3 
72.4.2 15, 29 oe 
; y & x y Y 
a 22-2 x+y 
8. P| a 16. 7-y oo 
1 e+ 2Qry+y? a®+ 2ab + 0B 
1. @ias6t: pee 
18 (a+1)(@+2) @+2)@+3) 1 
, a+3 a+1 a+2 


ART. 96 | 


19. 
20. 
at, 
ye 
23: 
(m5) (m+5) 


MISCELLANEOUS EXERCISES 


2 


e 


EXERCISES 147 


ax + 2? x? — a? 

ax — a? 2? + 83ax + 2a? 

e+e ry—x yt 

yp 0 ea 
2_ 1] a+ 4a+a? 

es oe 

















BE Ys my: 2 
ar (x? — Qry + y?). 
(ate b \. 
a+b b a+b/ 


In the following Exercises, 1—6, the letters are assumed to 
‘be positive integers. 


By Hich is larger, = or =? Why? * or 2p Why? 


: 


Le 


im 8. 


or 
earn 


Which is larger, or? Why? zor a Why? 


Which is larger 
a 
Which is larger, = or 


» Which is larger, : or 


OL 


or 


1 —1 
ee pla 
ee Explain. 





+ Ls 
a a! Explain. 





3 
9 
ane Explain. 


Which is larger, y or 9, 
zoey 


a 


Does 
x 


of x and a? 


gq + 
r+3 


. Does = = = ae all values of x and a? 


: for all values of x and a; for any values 





148 FRACTIONS [Cuap. XVI. 


x 











9. Is it true that ae = ? 
eg! bine ae re) 
8a _ ; 
ee 2 : 
10. Does [oy Dae Explain. 
a i A)» 
ala Does erga TR Explain. 


12. Which will give the greater product, if x and y are posi- 





tive integers, to multiply a positive number by 3 or to multiply 


the same number by mt Explain. 


13. Arrange in order of size, if a and x are positive, (a) if x 
is greater than 1; (b) if x is less than 1. 
a a a a 
| e etl 243 2 
14. Fill the blanks in the following : 























oh ins (gO ea , 
b hab 9) ae 
(b) le 2s 1p 2 2 Saas 
c-y 2£-y 1-2 2-2 eee Loe | 
x ? ? 


) @-D(b-) ~ ©=a) C-8) @—-H)-H) 
15. Express 2) 8) 7 as fractions with denominator 24. 


16. Express = m : as fractions with denominator abe. 


17. Express — as fractions with denominator 2Aabe 


ia’ 6B Be 


18. ‘lixpress:'= 3 an Oa a as fractions with denominator 9a°t?, 
x+y x-—dsy 
2a — 2b 3a + 3b 






as fractions with denominator 





19. Express 
6(a? — 6). 


“Arts. 96, 97] | DIVISION OF FRACTIONS 149 
3a 4b é 
(a — b)? (a + b) (a—b) (a+b)? 


' Reduce to a single fraction and simplify (leaving the de- 
nominator factored) : 





20. Find the L.C. D. of and 









































| 1. 5+ oa. LO eer ea 
a2, Wo. Lupeeree yaar 
23. a ae ieee 
24. ay se sa eS 
25. “4S. a 
| 26. ee 32. i ee 
33. : 


@-2)@-9)' @-HW-4* W-) &-® 

97. Division of fractions. The method of dividing one 
fraction by another is the same in algebra as in arithmetic. 
By the definition of division, the divisor times the quotient gives 
the dividend. 


Hence, to divide 3 by ? is to find a number ¢ such that 
3 2 


ees (1) 
Multiply both members of (1) by 4. This gives 
DAL eS. 


In general, to divide : by means to find a number gq such 


‘that 2) 
Cc a 2 
at 


150 FRACTIONS [Cuar. XVI. . 


To find g, multiply both members of (2) by : - This gives 
ad) ode (3) 
Sha on hee 
From (38) we note that the quotient is obtained by multiplying the 
dividend by the divisor inverted. 































EXERCISES 
Perform the indicated operations and simplify: ) | 
3 2 a a 
| 
12a—-6 2a-1 
Brake Pe 
ye ese Yh i. “ao ee 
3 BS: aOR 12 _4 40 4Ge eta 2 
DGS ipoaeee ’ a(a?-—a—12) 30° tom 
4, oath , 4c 43 21 
GeryO owe cOGs ° gt — OP 2+ 3y 
5, Oe. fey. i 2 tee 
62° Aa * oS 1 44 ieee 
6 (Sy + (Bab)? 1s, Ue. ee 
SANE (26) ° g— yay? 33 ey re 
Gar eevee ab = fa+b b ) 
7 (F+4) ; 16, + ; ot or 
a A + 
8. (5 ‘) yh (m— =) + (m+=) 
Caec e e—x—-6 2-2-3 
Ged ue ve—x—-2 w#+2-2 














Arts. 97, 98] COMPLEX - FRACTIONS 151 


Tere it si 


Be fang) *(*-2 53) 
Cra Bb + 


4ab(a + ti 20) 
a—b 


G+ 9)-G-1-4)] [0-9] 


1 ten Vee 2 ao 
25. }—+—4+-—]+]/-4+-4+°-}]. 
eee Tee” 42 ae wifin 


98. Complex fractions. A complex fraction is an indicated 
division in which either one or both terms are fractions or con- 
tain fractions. A complex’ fraction can be reduced to a simple 
fraction. 














2 
23. (a 2 prs 2a? + 2b(b + Sin Sa 


2a*7-—3ab+P 





Example 1. Reduce = to a simple fraction. 


oma polco 


SoLtuTIoNn: Multiply both terms by 10, the L. C. D. of the numerator 
and denominator. Then 








eS ee 15, 
SORT eae 
243 
Example 2. Reduce =~ to a Sauls fraction. 


ne 


Soxvution: Multiply both terms by 3zy, the L. C. D. of the numerator 
and denominator. Then 


24 szy(2 ae 








3a 


FRACTIONS FCuar, XVM 


EXERCISES 


Reduce the following complex fractions to simple fractions: 
























































23 ae Lp | 
1. 32 at 
ul 10. 5 = 
9 3,+4 ke 
"244 Z 
a? — 6? 
4 +27 a 
3. ah 11 ae 
2a 
1 m—n 
toe. +1 
4. <. 12) ae 
Fie! i = 
(ys m+n 
1 man, mtn 
5 0 13 mtn m—nN 
; b 1 he 
a m+n m—n 
1 
aaewit 14, —— 
ge Sh sea 
al ar x—2+- 
ae 15. 
i 1? 
sata Lee 
x Cee 
eb 
8. areas xy 
ees 16. 4: 
a z+e 
rhode i 
9 ee 17 1 an 
a b * tere hae 
baa 1m m 





- Art. 98] MISCELLANEOUS EXERCISES 153 





























1 2 
@+1) +25, ate 
), eae oes ie 
ae tia, 
242 meat i 
2 3 
: cee ae 
eee og rp peas ) 
19. : r+5 
b eee 
b ered ta 
3 5 
1 1 1 rf 
eae ; aang oer a+b b2 
1 + ; peat 
1 +a 
G0 
24° 2u.-- Z 
or -b 
Peale 
AG 


MISCELLANEOUS EXERCISES 


1. Which is the greater —) when xand y are positive 


numbers? 
2. Which will give the greater quotient, if x ia y are posi- 


j 55 or 3 


tive numbers, to divide a positive number by i or to divide 
the same number by mi Explain. 


: : 22 
3. Fill the blanks in the following: ang pena | sy 
Seas Cay 
wy(2 + 2?) 
4. The reciprocal of a number « is That is, the recipro- 


eal of a number is 1 divided by that number. What is the 
product of a number and its reciprocal? 







154 FRACTIONS [Cuar. XVI « 


5. From the definition of ep in re ; give the 
reciprocals of the following: (a) 2; (b) ® 43 (c Ne i  (d) = ae ; (0) 5: 


6. Show that to divide by a rane we Eee simply 
multiply by its reciprocal. | 


Perform indicated operations and simplify : 
3a a geet 1 


(Serre eae 13. 








4x + 2 _ 2a +1 























. 3 
3 mits = ST 15. ne 
2 
pe va x 4+ ax 16 aa 
e ti Pi a _- <4) es 
(x — a) x—a 1 erty? 
x+y «u-y 
1 eee 
11 . re pppoe 
“ath _2+y ey 
3 ———_——_ 
r-y «x+y 
Ye 











ee 
3 eee | 


























Ox 
19. (eo ~ 1) zs (sey — z = . 
a2 ete *) 
ae a 


“Art. 98] EXERCISES 


— 


R 


23. 
24. 
25. 
26. 
27. 
28. 


29. 


& 


m2 
m—n? 1 
wn 
n 05 
ax — bx + ay — by 
axy — bay 
6xyz + 3x72 + By" 
2x72 + 4aryz + 2y?z2 
ax — bu + cx + ay — by + cy 
axy — bry + cry 
Sayz — l5yz + 10xz — 302 
5a’z — daz — 302 
4ax* + l4ax — 30a 
62? + 21a? — 45x 
ve 4s 
32° — 6x2? + 12x 
Gaz? — ax? — 12ax__ 
6ax* — 17ax*? + 12ax 








(mate iat ee Tes mat), 


m n m+n 


(e+) e-)- Gra) 


32. 


33. 


34. 


oe 


or paul | res Ss oe a GR 


\m-n m+n 


w@—xr-—6 2-2r-3 2 -2r4+1 


ar ey 7 tr 2 


e@+2e+4 +8 wv 4 





y? — 9 


+2 y= 8 442 


(a +6)? a 


a Aa’ bi 





ac—a*—ab (a—c)?—8 be 


—ab +b? 


155 


47. 


48. 


Ge 4129) 


(aly eo Pe a ae 
dan Aelia eee 


Cine COON (L — q)* 
; iGsaed (y — a) 


[op Orn 1 ee ee 


x 1 3 
'@-0@_- @-24-2) 12 aseee 
1 lo 1 
Baa eae 
I pat Loe ne aa 3 ; 
ey at ay — 2y? 9 4 Baye 
a 2(1 — a) l—a 


FRACTIONS [Cuar. XVI. 




















x+1 = x+4 
(w — 1) (2-2) 











a+y a-% (4-2) (a—-y) — 

1 1 1 
Gob) G=0 wehbe 

1 1 1 1 1 1 




















2 3 f 














+3442 708+44+3  atasbomee 





iy 
tt 
i) 


“Ant. 98] REVIEW EXERCISES 157 
REVIEW EXERCISES 


1. Give an example of a linear equation in an unknown x ;1n an un- 
known y; in an unknown t. 
2. State whether, according to the definition of a factor, (a) } is a 


factor of 4; (b) * is a factor of x7; (c) mn is a factor of mn. Name all the 


factors of az?; of 12. 


Find the following products. 


Bea — 3)(a + 7). 7.. (% +0). 11. (60 + 7)3, 
tr 1)(27— 4). 8. (0 — x)? 125 bere 
5 Qr+3)Qr-3). 9 (4045) 13. (4h + 6¢ + 8)2. 
, 6. (4¢ + w)?. 10. (70 +1)(70 -1). 14. (82 +7)(4e — 5), 


15. Tell which of the following are rational and integral with respect 
tor: 3a2x3; a + mr +53 5V/z +17; 14Vazr + 2°. 
16. Tell which of the following are not trinomial squares and give rea- 
‘sons: (a) x* + 2ax +a’; (b) m? — 2mn — n?; (c) a? —ab +b?; (d) 4 +8b + 16b2. 
| 17. Describe the way in which the terms of a trinomial square are 


made up. 
Combine each of the following into a single fraction: 

















Es aS ry Weg te 
a -a ab a+b rss qr 
1 won, 24 o : 
19. +2 22, 5 45 25. 5 -- +2 
Aj ae age 26, ~ _¥. 
yr a+b inet 
Simplify the following: 
b 1 4a 3b 1 
27. a-7- 30. (a+1)-—. apie $6.5 
ot) a a V4 ORAS Ge. oa al ear 
b x a 2 ee x 
al - 
20. =-10c, 92. °F = 12py 35, 2 +3 Los 
5 52 Zi 38 a Or. 





9 








= | 
$ 


158 FRACTIONS [Cuap. XVI-%} 
39. Find the remainder when the sum of the squares of a and 6 is_ 


taken from the square of the sum of a and b. 
40. Find the remainder when the difference of the cubes of a and b- 
is subtracted from the cube of the difference between a and 6. 


Perform the following additions and subtractions: 


Oia ak a=—o 43 a b 


oS gh Ro 0 eS ‘ @+2ab a +4ab + 4b 


x7 2x a-2x b+e 
os 6a? 452-4 10x*4+ 7x —6 a az +bon +exr (a+b? —-2 


Find two solutions for each of the following equations: 











45. x? — 62 = 27. AY. 32 —4z =a. 
46. 427 —- 16 =0. 48. 32727 + 7x = 6. 
1 —=2 1-2 qg= 17 =f 
49. Add Tre ees Fete: Wek yp 3 


1 vito 3 
50. Add G— end)’ C-Had--) @-)e=e) 





CHAPTER XVII 
FRACTIONAL AND LITERAL EQUATIONS 


| 99. Clearing equations of fractions. We have solved 
‘(Chapter IX) some equations in which fractions are involved. 
-It is often convenient to clear such an equation of fractions by 
‘multiplying each member by the L. C. M. of the denominators. 





= To illustrate, solve the equation 32 a 3(@-1) 99 


2 baie #110) @) 
SOLUTION : 
Multiply each member by 10, the L. C. M. of 2, 5, and 10. 
This gives 15z + 6(2 — 1) = 99. (2) 
IPransposing and collecting, 21%°= 105, (3) 
—_ 
3 3(5 —1 L5- 2127. 99 
CHECK: Bo a 224? _w 





EXERCISES AND PROBLEMS 


Solve the following equations and verify the results by 
substitution: 


1. §@—2-t§ =44(8e4+1). 6. 2(2 +1) 4+ 52 = 46. 























47 4 By = 42 a: 
B 2. $2 + gr = 4g. 6. 2p 2 a8 _5 
3 r+3 2r+1_. 

ee 9 7, 4 Wwt+6 

4x+1 1 epee 7 
| 4, = feeeee = 4 

Direei--3 1 7 

5 Ga ts(*-3) +a ~ 9 
| ae oe DY = Se 
Cas bear’ 10 Sire eae Ce 


160 FRACTIONAL AND LITERAL EQUATIONS [Cuapr. XVII. | 


11. («© + 3)(2x + 1) = (@ + $)(2e — 3). 

12. (ce -—3? -@+9)@—® =O. 

13. (x — 2)(a — 3) — x(a — 4) = 9. 

e-1 a¢=—2- Visi 
7.3) (2 

15. (x — 3)(a+ 4) — 2° 4 Or = 3. 


14. 








100. Unknowns in the denominator. In some equations 
the unknown appears in a denominator or in both numerator 
and denominator. In these equations as in the equations of 
Art. 99, the first step in the solution is clearing the equation 
of fractions by multiplying each member by the L. C. M. of 
the denominators. Why the L. C. M. is used rather than any 
common multiple of the denominators, will be taken up in the 
second course. 


Example. Solve 22-1 3-2) 
22 — 3) Sa 





SouuTion: The L. C. M. of the denominators is 2(x — 3) (8x — 1). 
Multiplying both members of the equation by this expression we obtain - 


(2x —1)(82 —1) = 6(% — 3)(@@ — 2), 


or 62? — 5a +1 = 62? — 302 + 36. 
Then 252 = 35, 
and | x= f. 


Substituting x = “ in the original equation, we find the equation satisfied. 








a 
EXERCISES 
10 4 aera, 2 at ae 
TEN EVP Gy 1. 73 eee 
Pion yo ee gee) 
3 ei epee 
Mie a erage 7 


PRE par SYST 2” F(a +3) 


Arr. 100] EXERCISES AND PROBLEMS 161 





















































2 ae 
227 = 1 4 ar 
a z—1 ens | 
7, 3@+4)_ 3Qe-1)_, 
Seen 5 el ee eae 
, ea ae 
en ie een Gee 
Mee tT 1 
eo Gh oe 4t2 2+ 
| 4 3 3 4 
ee ss ia 
4 9 6 1 
are ny I ae fe mee 
6 9 4 1 
Se ets pee ae fs 
13. 2 3 4 11 


eter 2.748) 7a +4) 
PROBLEMS 


1. One-half of a certain number plus one-third of the 
numberis10. Find the number. 

2. The sum of two numbers is 66. One-half of the smaller 
plus one-seventh of the greater is 18. Find the numbers. 

3. The difference between one-third of a number and one- 
fifth of the number is 6. Find the number. 

4. The sum of a certain number, its half, its third, its 
‘ourth, and its fifth is 274. What is the number? 

5. What number must be added to the numerator of 
#7 in order that the resulting fraction shall be equal to 4? 
6. What number must be added to the denominator of 
*y in order that the resulting fraction shall be equal to 4? 


162 FRACTIONAL AND LITERAL EQUATIONS [Cuap. XVII. 


7. What number must be added to both numerator and 
denominator of 38; in order that the resulting fraction shall 
be equal to #? 

8. A yardstick is cut into two pieces. One piece is ¥ 
the length of the other. What is the length of each piece? 

9. What number added to both numerator and denom- 
inator of the fraction 2 will double the value of the fraction? 
What number will halve the value of the fraction? 

10. Find two numbers which differ by 4, and such that 
one-half the greater exceeds one-sixth of the lesser by 8. 

11. $1000 is divided between A and B in the proportion 
3 to 8. How much more is B’s share than A’s? 

12. A man leaves one-third of his property to his wife, one- 
fifth to each of his three children, and the remainder, which was 
$1200, to other relatives. What was the value of his estate? 

13. An estate of $5300 was left to two heirs. The first 
received one-third more than the second and $400 additional. 
How was the estate divided? 

14. A man has two sums of money at interest, which ton 
gether amount to $19,000. One sum brings 5% interest, the 
other 3%. From the latter he receives $250 more income than 
from the former. What is the amount of each of the two sums? 
7 15. The width of a room is two-thirds of its length. If 
“the width were five feet more and the length 8 feet less the 
room would be square. What are the dimensions of the floor? 

16. Each year a merchant increases his capital one-third, 
but takes away $4000 for expenses. At the end of the second 
year, after deducting the second $4000 he finds that his capital 
has increased one-half. What was his capital when he began 
ae | 

7 . The denominator of a fraction exceeds its numerator 
by 3 If 1 is added to both numerator and denominator the 
resulting fraction will be equal to 3. What is the fraction? 

18. A lazy boy is asked to ide one-half of a certain. 
number by 8, and the other half by 10, and add the quotients. 


F Anv. 100] PROBLEMS 163 


_ To save work, he divided the number itself by 9, and obtained 
~aresult too small by one. Determine the number. 

19. At the time of their marriage a man’s age was to that 
of his wife as 3 is to 2. Nine years later, it was as 4 is to 3. 
What was the age of each at the time of their marriage? 


Hint: Let 3x = age of the man at marriage. 


20. A tank is emptied by two pipes. One can empty it in 
30 minutes, the other in 25 minutes. In what time can the 
_two together empty it? : 

| 21. A can do a piece of work in 3 days, and B can do it in 
Od days. In what time can they do it together? 


Hint: Let x = the number of days it will take A and B together, then 


. x = the part A and B can do in one day. 


| 22. If A can do a piece of work in 8 days and B in 10 days, 
in what time can they do it together? 

23. Smith and Jones have forgotten the scores in the foot- 
ball game between Chicago and Illinois in the fall of 1914. 
Jones remembers that the difference of the scores was 14, and 
Smith remembers that the difference of the scores was half as 
‘much as the sum of the scores. What’ were the scores of the 
two teams? : 

24. Smith and Jones have forgotten the scores of the foot- 
ball game between Chicago and Illinois in the fall of 1913. 
Smith remembers that Chicago won by 16 points, and Jones 
remembers that Illinois’s score was 2 less than half that of 
Chicago. What were the scores? 

25. A man is now 45 years ‘old and his son is 15. How 
many years must elapse before their ages will be as 13 is to 7? 

26. Find a fraction whose numerator is greater by 3 than 
half of its denominator, and which when reduced to its lowest 
terms is 2. 

27. Find the age at which the Greek mathematician Dio- 
phantus died. His epitaph reads as follows: Diophantus 


164 FRACTIONAL AND LITERAL EQUATIONS [Caap. XVII. « 


passed + of his life in childhood ; +; in youth and + more as a 
bachelor ; five years after his marriage was born a son who 
died four years before his father at half his father’s age.* 

101. Literal equations. A literal equation is one in which 
some or all of the known numbers are represented by letters. 
It has become customary to represent the unknowns by the | 
last letters of the alphabet, while the known numbers are 
usually represented by the first letters. 


Thus, the general linear equation (Art. 65) 
ax +b =0 
is a literal equation. Other examples of literal equations are 


—+b=6, (a—b)z=b+c, (8 —a) +2 +a 


EXERCISES 


Solve the following equations, the unknowns being repre- 
sented by the last letters of the alphabet: 








1. -4b=c. 
fe 
Sotution: Multiply both members by z, and obtain 
a+bx =cx. (1) 
Transposing and collecting terms, we have 
(6 —c)x = —a, 
and | dn a (2) 
CuHEcK: Substitute in (1), a+ c = xt < a 
ac —ab + ab = ac. 
020: 
2. (a—b)t=b+. 4, ee 
x x 
3. (8 —a) + (24+ ))r =. 5. ay +cy =2(a+b). || 


* Father’s age here means his age at his death, and not at time of his 
son’s death. 


\ 
4 


| 


A 
J 


\ 
t 
i 


“Arts. 101,102] | SUBSCRIPT NOTATION 165 






































6. az + bz = 3(a +b). 10. as —a® — 4 = 3a — sg. 
b 
i dx = 2 — d?. a= ———: 
eed 1 = C a sarees 
, 2 ae ip eee 
U U 2 
9. a(2a+x—1+d)=2b. 13 en ae Needs 
1 he be 
14. (a+ 2)(b—2x) —- (a—2)(b+2) = 2. 
c+z2z.c-2 1 
0 ce aa a naa 
r+a r iL a 
16. - 2a Ea 
a+b at—b by-a 
17. a+ ee ii ; Solve for y and then for ¢. 
18. = @ 19. = . 
2 2 3 
8 
i 
ay} 1 
Ne 8 Mee cor rar 
+b 
2—a 


102. Subscript notation. It is often convenient to repre- 
sent related numbers by the same letter with small numbers 
(called subscripts) written at the right and below the letter. 
Thus, if 7 represents temperature, the temperature of a body 
pt two different times may be represented by 7), 72. The 

weights of three bodies may be represented by Wi, Wo, Ws. 
These letters are read “7 sub 1,” “T sub 2,” “W sub 1,” and 
Bo on. 

Another notation less often used is the prime notation. 

Two or more letters are distinguished from one another by 


166 FRACTIONAL AND LITERAL EQUATIONS [Cuar. XVIL 


marks (called primes) placed at the upper right hand. ‘Thus, 
temperatures at two times may be represented by 7’, T", and 
weights of different bodies by W’, W”, and W’”. ‘These are 
read “7 prime,” ‘7 second,” “ W prime,” ‘‘W second,” and 
“W third.” 
Both the subscript and prime notation are used in physical 
and engineering formulas. | 


TRANSFORMATION OF PHYSICAL FORMULAS AND PROBLEMS 


In the following physical formulas, solve for the letter 
indicated. The custom of using the first and last letters of 
the alphabet for the knowns and unknowns respectively is not 
followed here. 
1+2 
2. H = 1082 + .3805t. Solve for é. | 
8. C = 8(F — 32). Solve for F. | 


1. h= 


Solve for 2. 


én 





4 = - Solve for n. 
R+o 
5. C= eae uae Solve for r. 
R19 
W+R 
6. Q = W+tRtw Solve for R. 


W'(H — t+ 32) 





1... Wi 966 Solve for ft. 
+ 7d ; 
8s. s7hitiene Br Solve for d and for @. 
W —- W'’ 
9. A= H Wiw+w” Solve for W. 1 
10. F = ( : , (",) Solve for 7’. j 
Vy ae 5 


/ 





| Arr. 102] EXERCISES AND PROBLEMS 167 


Read aloud the following formulas, then solve for the letter 
indicated. 


a Fu fe C(ts a t1) i W(t, — t1) = W a(t — ts). Solve for ts. 
12. Vi; = Vo(1 + .0087¢,). Solve for ,. 





(i) a Ba 
Ct See . 
3 Big St, Solve for 7; 
0, ~0;) = Ee Baise fan re 
1 
7.65 765 : 
Laide Bee T pr Solve for 7”. 


16. A = $bh” + 4a(h” +h’) + 4ch’. Solve for h’. 


V1 

/ | oes 

” P v2 

17. PP” = — Solve for ve. 

pier 2 

202 
18. If a and b are the altitude and base of a triangle and A 

the area, then A = 2. Express a in terms of 6 and A. 

19. If r per cent is gained on an article that costs ¢ dollars, 


the selling price S is given by the formula S = c + in 
Find c if r = 10 and the selling price is $181.50. 

20. Write a formula for the selling price when r per cent is 
lost. Find r when the selling price of the article in Problem 
19 is $138.60. 

21. The proceeds P of an amount of a dollars discounted 
for ¢ days at r per cent a day is P=a — sa00" Solve the 


equation for a; for r; for ¢. 


22. If a cents be divided between two boys so that one 
boy has 6 cents more than the other, how many cents has each? 


168 FRACTIONAL AND LITERAL EQUATIONS [Cuap. XVII. 


23. The pressure of water per square inch at a depth d feet 
cae 2. nS 
is given by the formula P = ad a d. At what depth is the 
pressure ten times as great as the pressure of the air at the 
surface? (Pressure of the air to be taken as 15 pounds per 
square inch.) 

24. If two quantities of water mm and m, at temperatures 
t; and tf. are mixed, the temperature, ¢, of the mixture is 


7 mt + Mato 
M1 +- Meg 


How much water of temperature 48° must be mixed with 6 
gallons at 150° to make a mixture at 110°? 


CHAPTER XVIII 
RATIO, PROPORTION, AND VARIATION 


103. Ratio. The ratio of a number a to a second number 


6 is the quotient obtained by dividing a by b. The ratio of a 


- to 6 is usually written in the fractional form “, but the form 


> b 


) a:6 is often found. All ratios of numbers may be considered 
_as fractions. 


The two numbers a and b in a ratio are called the terms 


of the ratio. The numerator is called the antecedent and the 





denominator the consequent. 


EXERCISES 


In the following exercises name the antecedent and the 
consequent for each ratio; write the ratio in fractional form 


and simplify by reducing the fraction to its lowest terms: 


Ratio of 4 to 12. 

Ratio of 12 to 4. 

Ratio of 33; to 6. 

Ratio of 74 to 7%. 

Ratio of 3 minutes to 2 minutes. 
Ratio of 3 minutes to 2 seconds. 

Ratio of x? to xy. 

Ratio of a2 —b toa+ 0b. 

Ratio of 16 pounds to 26 pounds. 
Ratio of 16 pounds to 26 ounces. 


Sager) Bahai tapeiles pte oo SEs 


= 
a 
Kalco 
ch 


— 
iS) 
8 

coo 
xR 


169 


170 RATIO, PROPORTION, VARIATION  [Cuap. XVIII.’ 
13. (1 _ “) : (1 a “|. 
y y 
y" 
14. (u+2 U) (4° -¥). 


104. Proportion. A proportion is an we equality of. 


two ratios. If the two equal ratios are and - then the 


proportion is 


and a, b, c, d are said to be in proportion. The proportion is 
often written | 
C50 00 

An older notation sometimes found is a:b::e:d. In any 

notation the proportion may be read “‘a is to 6 as ¢ is to d.” | 
The first and fourth terms of a proportion are called the 

extremes, and the second and third terms the means. ‘Thus, 

in the proportion, 


a and d are the extremes and 6 and c the means. 

Since a proportion is an equality, operations that may be 
performed upon the two members of an equality may be per- 
formed upon the+two ratios of a proportion. Thus, in the 
proportion, 


both members may be multiplied by bd, giving 
ad = be, 


which proves the theorem: 
In any proportion the product eo the means equals the product 


of the extremes. 





Ht 
' Arrs. 104, 105,106] MEAN PROPORTIONAL 171 
EXERCISES 


In the following proportions test the above theorem: 








eis Pics 
6. 35 ' 137 408 
ee a 85a 
Bri 1 " 17y = 289xy 
a 
5 ax+ay  _b 
' bn —~ by a-—y 
r+y 


Find the value of x in the following proportions: 


_ ee 11. a:b=Z:¢. 

5 ee 
ees 1 ety Cz: 

x o 
8) 73:07 = 3:8. 1385 377-2 = (ix +3. 
g, =:3) = 3:8. 14. —-2:7-4=3:2742. 
104).0 = C > 2. 15. +:40 = 100 7:4.7.- 


1065. Mean proportional. If the means of a proportion 
are the same number, this number is called a mean propor- 
tional between the two extremes. Thus, in the proportion, 
2:6 = 6:18, 6 is a mean proportional between 2 and 18. In 
the proportion a:z = «:b, x is a mean proportional between 
a and b, and from this we obtain 


x= + V/ qb, 


106. Third and fourth proportional. A third propor- 
tional to two numbers, a and 6), is the number z, such that 


b 


x 


be 
b 


172 RATIO, PROPORTION, VARIATION [Cuar. XVIII. | 


Thus in 3°; = 33, 48 is a third proportional to 3 and 12. 
A fourth proportional to three numbers a, 6, and c taken 
in the order given is the number z if 


Thus in 3 = 45°, 9 is a fourth proportional to 5, 3 and 15. 


EXERCISES 


Find the mean proportionals between the following pairs 
of numbers: 





1 1 
1. 2 and 8. 5. F and 50° 
2. 4 and 9. 6. 2 and y. 
3.. 2 and 32. if 3 and = 
xe ci 
1 
4. 1 and 64. 8. = and 2. 
Find third proportionals to the following pairs of numbers : 
9. 2 and 4. 12. (a + b) and (a — BD). 
| by at gic 1 1 
10. 3 and : 13. e and ii 
11. Bog —1. hy = a 3 
7 u—y y 


Find fourth proportionals to the following sets of numbers 
in the order given: 


ee SOAS 18. %¢, Daa: 

16s), 2) 19. a+b, ab, a — b. 

17. a, b,c. 20.\2;° Sie 
Cee 





| Ars. 107, 108, 109] PROPORTION 173 






10%. Proportion by alternation. If, in the proportion 


eo fs 
eo 
_ both members of the equation are multiplied by °, the equation 


becomes after reduction 


That is, in a proportion the means may be interchanged with- 
out destroying the equality. In this case the second propor- 
tion is said to be obtained from the first by alternation. 

| 

j 

| 


108. Proportion by inversion. From the proportion, 


we have from Art. 104, (1) 
be = ad. 


| Dividing both members of this equality by ac, we have 
i b d 
ae: 2) 


_ It may be noted that the members of (2) are obtained from (1) 
by inverting the fractions. For this reason the second propor- 
tion is said to be obtained from the first by inversion. 





109. Proportion by composition. If, in the proportion, 


a c¢ 
| ees (1) 
1 be added to both members, we find 
| a tr a+b cec+d 
fee ot 1 or =) eae ee (2) 


| In this case, the second proportion is said to be obtained from 
the first by composition. 


174 RATIO, PROPORTION, VARIATION [Cuar. XVIII. | 


110. Proportion by division. If 1 be subtracted from both 
members of the proportion, 


there results ah ts 


and the second proportion is said to be obtained from the 
first by division. 

111. Proportion by composition and division. From the 
proportions by composition and by division we obtain, after 
dividing members, 


a+b ct+d 

a—b c=—d 
This proportion is said to be obtained from the proportion 
: ae by composition and division. 


Composition, division, and composition and division are often very 
appropriately called addition, substraction, and addition and substraction 
respectively. 


EXERCISES 


Write proportions obtained from the following by (a) 
alternation, (b) inversion, (c) composition, (d) division, (é) 
composition and division: . 


5. 3:4=(e+y):(@-y). 


From each of the following equated products obtain five 
proportions: ; 


24 2 4 
6. 2-10 = 4-5. 8 5 me 
7 ¢-3= 21-1; 9. 2-(4+y) =3:(z—-y) 


Pieter cs. 


“Ants. 111, 112, 113] VARIABLES AND CONSTANTS 175 


10. 3:4=24-1. Pas *a1- do = b1 « be. 
1h A 3 = 6. Vos fe PT» = 1° XV. 
14. met prove ~~ 4 _ oe. 


Ma — a | by — by 
eae iy no iss 


16. is = 4S prove X1 + 322 M Yi + BY2_ 
Y2 v2 Y2 


ay by 
Te it ay by’ Prove 





! Find the value of x in order that the following proportions 
: 


i eW+1):@—1)=2:3. 19. @+2): (8244) =5:6. 


/ 





fete 1)s (4 +2) =2:3. 20. 





| 112. Variables and constants. If ¢ represents the time 
“since a Chicago train left New York and s its distance from 
|New York, we note that each of these symbols s and ¢ takes 
‘different values during the progress of the train. On_ this 
‘account, ¢ and s are said to be variables. If the train should 
Tun at a uniform speed of v miles per hour, we know that 





Sk 


| In general, a symbol is said to be a variable if it may repre- 
sent different numbers in a discussion or problem. It is con- 
stant if it represents only one number. The idea of a variable 
‘plays an important réle in algebra (Chap. XIX). 
113. Direct variation. When two variables are so related 
‘that their ratio is a constant, either one is said to vary as or to 
lat directly as the other. 


In symbols, i Sakon ker. 
when & is a constant, may be read ‘‘y varies as 2.” 
| * See Art. 102. 





176 RATIO, PROPORTION, VARIATION [Cuap. XVIII. . 
¥ 


In describing the progress of the train, (Art. 112) we write 
! s = ut, 


and say that the distance from New York varies as the time 
since leaving New York. Suppose v = 40 miles per hour, then 


s = 400. 


EXERCISES 


Write in the form of an equation each of the following 
statements : | 
1. The weight W of the water in a tank varies as the 
volume V of the water. 
2. The simple interest earned on a principal P varies as 
the time ¢. 
3. The speed v of a falling body, started from rest, varies 
as the time ¢ since it began to fall. 
4. If y varies as x, and « = 2 when y = 6, find y when 
en), 


SoLution: The statement y varies as 2 means 


y =kx (k constant). (1) 
To determine k, make x=2,y =6. 
This gives 6 = 2k, (2) 
or k =3. (3) 
From (1) and (8), y= Sf. (4) 
Substituting, y = 30. 


5. If y varies as x, and x = 4 when y = 12, find y when 
A 0; 

6. The area of a circle varies as the square of its radi 
If a circle whose radius is 10 feet contains 314.16 square feet, 
find the area of a circle whose radius is 8 feet. a 

7. In using a spring balance, the principle applied is that 
the stretch s of the spring varies as the weight W to be de- 
termined. A weight of 10 pounds stretches a certain spring 
1 inch, how great a weight is required to stretch it 2% inches? 


Arr, 113] EXERCISES AND PROBLEMS 177 
8. The distance through which a body falls from rest 
“varies as the square of the time in seconds. If a body falls 
16 feet the first second, how far does it fall in 10 seconds? 
9. The weight of a wire varies as its length. It is found 
that 100 feet of a certain wire weighs 3 pounds. Find the 
weight of one mile (5280 feet) of this wire. 


EXERCISES AND PROBLEMS 


Find a mean proportional between 3 and 48. 
Find a fourth proportional to 5, 6, and 35. 
Find z in the proportion 50:75 = 90: x. 
In a certain business transaction, A gains $200 and B 
_ loses $50, and then A’s capital is to B’s capital as 4 tol. If 
_A’s original capital was $1200, what was B’s? 

5. Find two numbers, one being twice the other, such that 
their sum is to their difference as 5:1. 

6. How may $10 be divided among three boys so that 
for every dollar the first receives, the second shall receive 15 

cents and the third 10 cents? 

7. The sides of a rectangle are in the ratio 2 to 3. Find 
the ratio of the area of a square of the same perimeter to the 
area of this rectangle. 

8. In the state of Illinois in the 1910 census, the ratio of 
foreign-born white inhabitants to native-born white was ap- 
proximately 27.8 per cent. What number of each if the total 
white population is 5,527,000? 

. 9. A business worth $37,000 is owned by three men. The 
‘share of the one man is a mean proportional between the shares 
of the other two. If his share is $12,000, what is the share of 
each of the other two? 
10. Prove that no four consecutive integers such as n, 
n+1,n+2,n +3 can form a proportion. 
11. The Ist, 3rd, and 4th terms of a proportion are x + y, 
x — y, and (x + y)?; required the 2nd term. 


ye oo pO be 


q 


178 RATIO, PROPORTION, VARIATION  [Cuapr. XVIII.” 


12. If four numbers 2, y, 2, and w, not all equal, are in 
proportion, show that no number different from zero can be 
added to each which will leave the resulting four numbers in 
proportion. 7 

13. If ax — by = cy — dz, find the ratio of x to y in terms: 
of a, b, c, and d. 

14. Divide 44 into two parts such that the less, increased 
by one, shall be to the greater, decreased by one, as 5 is to 6. 


APPLICATIONS TO PROBLEMS FROM MENSURATION 


Two triangles that have the same shape are said to be similar. In 
two similar triangles the sides of the one taken in any order are proportional 
to the sides of the other taken in the same order. 


15. The sides of a triangle are 12, 15, and 20. In a similar 
triangle, the side corresponding to 


12 is 18. Find the other sides. 
oN Hint: Let 15z = one required side — 

and 20x the other. . 
His 16. The sides of a triangle are 


12, 15, and 20. The perimeter of a similar triangle is 188 feet. 
Find its sides. 


The areas of similar triangles are in the same ratio as the squares of 
the lengths of corresponding sides. 


17. A triangular field has sides 30, 40, 50 rods. Find the 
sides of a similar field of four times the area. 

18. A triangular field has an area of 20 acres. What is 
the area of a similar field with twice the perimeter? 4 

19. A triangular lot has one side 10 rods and has an area 
of 40 square rods. What is the area of a similar lot whose 
corresponding side is 25 rods? ; 

20. The areas of two similar triangles are 121 and 144, 
respectively. One side of the one triangle is 22. Find the 
corresponding side of the other. 


>, 


4 


& Arr. 113] EXERCISES AND PROBLEMS 179 


La 


ee ie wae 


Bie A horse tied with a rope 40 feet long in the center of a 
pasture eats all the grass within reach in 4 days. If the rope 
were 20 feet longer, how many days would it take him to eat 


all the grass within reach? 


Hint: The area of a circle varies as the square of the radius. 


22. The volume V of a sphere varies as the cube of the 
radius r, and the volume of a sphere of radius 10 inches is 4189 
cubic inches. Find the volume of a sphere of radius 8 inches. 


EXERCISES IN THE NOTATION OF PHYSICS * 


23. The ratio of the force F to mass m is a number k. 
Express each of. the letters /, m, and k in terms of the other 
letters. | 

24. The product of the pressure p by the volume v of a 
gas divided by its temperature 7 is a number k. Express each 
of the letters p, v, and 7’ in terms of other letters. 


25. Given horn uae solve for R, h, and r in turn. 





eo: Sar 
2 ay ats 
26. Given = = es solve for G. 
27. Given toes ok = solve for x. 


28. Suppose v varies as ¢ and v = 128 when ¢ = 4, find v 
when ¢ = 25. 

29. Given that s varies directly as the square of ¢ and that 
s = 64 when t = 2. Write this in the form of an equation, and 
find s when ¢ = 10. 

30. The force F acting on a body varies as the acceleration 


a produced in the motion. Write the relation between force 


and acceleration when the constant ratio is the mass m. 


* The teacher is not expected to take the time to explain the physical 
meaning of the relations given. The exercises are given to familiarize the 
student with the use of other letters than x and y for unknowns. 


CHAPTER XIX 


GRAPHICAL REPRESENTATION OF THE RELATION BETWEEN 
TWO VARIABLES 


114. Introduction. When the corresponding changes in 
two related quantities, as for example the temperature at 
successive hours of the day, are to be described, it is useful to 
represent the quantities by lines and points (See Art. 18). 
Such a representation is said to be graphical. By this method, 
the corresponding changes in the two quantities are presented 
to the eye in a very vivid way. 


For example, a daily paper gives the following temperatures for Chi- 
cago at successive hours of a certain November day : 


BRA Mi Soke 8 aM. 23° 2 P.M. ‘dog 8 P.M. 37° 
DA MeL 9 aM. 28° 3 P.M. 41° 9PM. 37° 
4 aM. 20° 10 am. 31° 4pm. 40° 10 p.m. 38° 
Db AM.. 20" 1 pW) es 5 p.m. 39° 11 p.m. 38° 
6 am. 21° Noon. 35° 6 p.m. 39° Midnight. 37° 
7AM. 22° 1pm. 38° 7 PM 38" 1am. 36° 


The changes in temperature with respect to time are readily grasped by 


the representation of these numbers on cross ruled paper (See Fig. 22). . 


115. Axes, Coérdinates. In graphical work, much use is 
made of two fixed perpendicular lines of reference. The lines 
of reference X’X and Y’Y (See Fig. 23) are called coérdinate 
axes and their intersection is called the origin. 

The horizontal line X’X is called the X-axis and Y’Y is 
called the Y-axis. The horizontal distance from the Y-axis 
to a point P is called the abscissa or x-value of the point. The 


vertical distance from the X-axis to P is called the ordinate or 
180 


ae ee 


/ Art. 115] . PLOTTING OF POINTS 181 


y-value of the point. The z-value and the y-value of a point 
are together called the coérdinates of the point. It is the 
custom to take distances measured to the right from Y’Y as 





































































; “Rees teaum cB Rie Wie | | 
‘ /< Sena mal 
\ » / Se ee Be Ra 
ee Re 
; a SePeer ATE 
; ot EERE 
: PERE 
tpt 
sreseiiatarts 
M. 1 


ye 
Fia. 23 





182 GRAPHICAL REPRESENTATION ([Cuap. XIX. “i 


positive and those to the left as negative; those measured 


upward from X’X as positive and those downward as 


negative. 


116. Plotting of points. If we have given two numbers, — 


say 2 and —5, we can find one and only one point that has the 
first number for its abscissa and the second for its ordinate. 
To find the location of the point for the numbers x = 2, y = —5, 


we start at the origin O and measure two units to the right 


along the X-axis, and from this point, we measure downward > 


a distance 5. This point may be represented by the symbol _. 
(2, —5). The symbol (a, b) denotes a point whose abscissa , 


is a and whose ordinate is b. When a point is located in the 
manner described above, it is said to be plotted. 


117%. Use of coordinate paper. In plotting points and 
obtaining the geometrical pictures we are about to make, it is 
convenient to use codrdinate paper. ‘This is paper ruled both 
horizontally and vertically as shown in Figs. 22 and 24. 


EXERCISES AND PROBLEMS 


1. Plot the points (3, 4), (3, —4), (—8, 4), (—38, —4). 

2. Draw the triangle whose vertices are (3, +1), (0, 5), 
(—4, —2). ) 

8. Draw the quadrilateral whose vertices are (2, —2), 
(—8, 4), (-6, —8), (3, 4). 


Historical note on graphical representation. The discovery of the 
method of representing functions and equations graphically is due to 
René Descartes (1596-1650), the French philosopher and mathematician. 
The discovery of this graphical representation of equations marks one of 
the greatest. advances ever made in mathematics. He showed that dis- 
tances measured in opposite directions could be used to represent positive 
and negative numbers, and through such representation brought mathe- 
maticians to see that negative numbers are indeed very real and useful. 


¥ 
q 


y 


: 


Z 


» Arrs. 117, 118, 119] VARIABLES 183 


4. If a point moves parallel to the X-axis, which of its 


codrdinates remains constant? 


5. If a point moves parallel to the Y-axis, which of its 
coordinates remains constant? 

6. A line joining two points is bisected at the origin. If 
the codrdinates of one end are (4, 5), what are the codrdinates 
of the other end? 

7. Draw the triangle whose vertices are (8,0), (0, 5), 
(—3, —2). 

8. Given a north and south line, and an east and west line 


, for reference lines (Y and X-axes respectively), the following 


fy 


coordinates of points on a river indicate its general course : 
(0, —1), cp —2), ee —23), (2, —13), (3, We (4, 5), (5, 10), 
(-1, 0), (—2, iy (—3, 2), (—35, 1), (—4, —1), (—5, —3). 


“Map the river from x = —5 to x = +9. 


118. Variables. A variable is defined in Art. 112. To 
illustrate again, if ¢ represents the time of day measured from 
2 a.m. (Fig. 22), and 7 represents the temperatures, we note 
that each of these symbols changes in value throughout the 
24hours. They are therefore variables. 

As a further illustration, we may think of a particle of 
matter whose position is given by (2, y) moving to various 
positions.in the plane of the codrdinate axes. As such a parti- 
cle moves along a curve x and y vary in value. 


119. Definition of a function. If two variables xz and y 
are so related that when a value of one is given, a correspond- 
ing value of the other is determined, the second variable 1s 
called a function of the first. Thus, the area of a square is a 


_ function of its side. This function may be expressed algebra- 


ically as 


— 72 
ete 


where z is the length of a side. 


184 RELATION BETWEEN TWO VARIABLES [Cuap. XIX. 


The simple interest, denoted by J, earned on a principal of 
$100 at a rate of 6 per cent per annum is a function of the time 
tin years. This function may be expressed algebraically by 


Leal. 


We have had many examples of functions. In particular, 
the idea of a function of 2 taking different values when x changes 
has been well illustrated in Arts. 18, 31, and 83. 


120. Graph of a function. By a method similar to that 
employed in Ex. 8, Art. 117, to map a river, a function may 
be represented with respect to codrdinate axes. ‘This repre- 
sentation of a function is called the graph of the function. 

Fig. 22 gives the graph of temperature 7’ as a function of the 
time ¢. 


Example. Obtain the graph of $v +4 for values of x between —5 


and +5. 
Let y = 34 +4. The object is to present a picture which will exhibit 


the values of y that correspond to assigned values of x. Any assigned 


value of x with the corresponding value of y determines a point whose 
abscissa is 2 and whose ordinate is y. 

Assigning values for « and computing the corresponding values for y, 
we obtain the following table : 


e|O|1 |2125 13 | 4] 5 |-1 |-15 | -2|-3 | -4| -5 
y14155|717.75|85| 10|115| 25 |-1.75] 1 |=0.5 ))—oq0—une 





and so on. The corresponding values of x and y are plotted as coérdi- 


nates of points in Fig. 24. 


It should be noted that there is no limit to the number of : 


corresponding values which we may compute and imagine 
plotted in a given interval along the X-axis, and further that 
to small changes in the values of x, there correspond small 
changes in the values of y. 


These facts suggest the idea of a continuous line or curve — 


to represent the function much as a continuous curve is used 
in mapping a river. The line in Fig. 24 is the graph of the 
function 3x + 4. 3 


F, oR) 


ba 


, Arr. 120] EXERCISES | 185 


































































































Two spaces = 1 unit 


Fic. 24 


EXERCISES 


Construct the graph of each of the following functions by 
plotting a number of points for each function, and drawing a 
continuous curve through these points: 


1. 32 +4. 4. 27 —1. 7. 62-1. 
Beer 4 6. 5.5 +5. 8. 7x +2. 


8. 4a — 3. 6. 4” + 2. 





186 RELATION BETWEEN TWO VARIABLES ([Cuaap. XIX. 


9. The temperature readings of a Fahrenheit thermometer 
that correspond to Centigrade readings are given by 
y = $x + 32°, where x refers to Centigrade readings and y to 
.Fahrenheit readings. Draw the graph to represent Fahren- 
heit readings to correspond to given Centigrade readings. 


10. A certain kind of cloth costs $2 per yard. What 
function gives the cost of any number of yards if c is the 
cost and n the number of yards? Plot the graph of this 
function. 


11. A man drives a car at the rate of 20 miles per hour. 
Write the function that gives the distance d he drives in ¢ hours, 
and construct its graph. 


12. Butter costs 30¢ a pound. Draw a graph to show the 
cost c of n pounds. 


121. Graph of an equation. If x and y are involved in an 
equation, say 
y—3¢-—4=0, (1) 


we often speak of the graph of the equation. If we express 
y in terms of x, we have 


y = 32 + 4, 


and may plot the graph of this function of z. The graph of this 
function is often spoken of as the graph of equation (1), since 


coordinates of points on the graph and of no other points satisfy — 


the equation. 

To construct the graph of an equation in xz and y, we have 
therefore merely to express y in terms of x and to construct the 
graph of the function thus obtained. 

The graph of an equation is perhaps better called the locus 
of the equation, since the codrdinates of points on the graph 
and of no other points satisfy the equation. 


bs 
\ 


Arts. 121,122, 128] GRAPHICAL SOLUTIONS 187 


EXERCISES 


Construct the locus of each of the following equations : 

1. y — da = 4. 4. 4x — 2y = 3. ~. ae2+y = 7. 

2.04 = © = 6. 5. o2-+ 4y = 9. 8. y — 32 =.2: 

3. 3x + 4y.= 5. 6. 2a Sy = 4. 9. 2y + da = 4. 
10. 62 — 3y = 3. 


122. Locus of a linear equation. An equation of the form 
az + by +c =0 


is said to be linear in x and y because its locus is a straight line.* 
To find the locus of a linear equation, it is only necessary 
to find two points of the locus and to draw a straight line 
through them. 
Thus, to find the locus of «+ y = 6, choose x = 0 and find y = 6; 


choose y = 0, and find « = 6. We may then locate the points (0, 6) and 
(6, 0) and draw a straight line through them. 


EXERCISES 


Draw the locus of each of the following equations : 
for oy = 11. 3. 4r—sy=90.. 5. 84 —y =3: 
ane Aly = 1. 4°62 + Sy = 4. 6. 6y — 2 ='8: 


7. Write an equation in which y increases as x increases, 
and plot its locus. 

8. Write an equation in which y decreases as x increases, 
and plot its locus. 


123. Graphical solution of equations. If we construct the 
loci of two equations, say of 3% — 2y = 6 and x + 2y = 10 as 
shown in Fig. 25, it is seen that the point of intersection of 
the loci is (4,3). As this point is on the locus of each equa- 


* No attempt is made here to prove this statement. The proof is 
given in analytic geometry. The fact is well illustrated in the above 
examples and may be taken for granted for the present. 


188 RELATION BETWEEN TWO VARIABLES [Caap. XIX. 
































Two spaces == 1 unit 





Fre; 25 


tion, the values x = 4, and y = 3 satisfy both equations. A 


pair of values, such as (4, 3) that satisfies each of two equa- 
tions is said to be a solution of the pair of equations. 


Exercise. Show that x = 4, y = 3 satisfies the given equations. 


Thus, the graphical solution of two equations is the point of 
intersection of the loci of the equations. 








Since the graph of a linear equation is a straight line, and ~ 


since two straight lines intersect in only one point, there is in 
general only one pair of values of x and y that satisfies a pair of 
linear equations in x and y. 


EXERCISES 


Plot the graphs of the following equations, and find solu- 
tion of each pair of equations from the graph. Test the solution 
by substitution in the equations. 


ey, 


»» Apts. 123, 124] GRAPHICAL REPRESENTATION 189 


1 xr+y =8, 7. 3x + 4y = 10, 
x—y =4. 5x — y = 9. 
2. x2-—y =4, 8. 5a — 4y = 11, 
2x + dy = 15. 4x + 2y =14. 
Baur —y =—.0, 9. 2x —y = 6, 
y+3z =0. 4x — 2y = 8. 
4,.x+y+6 =0, 10. 2x + y = 3, 

z—y = 0. 6a — 2y = 14. 
5. x — 2y =4, 11. 2x + y = 0, 

z+ 2y = 8. —y + 5x = 0. 
6. x+4y = 9, 125 37 Ay a7. 

ot — y= 14. 6x4 +y = 32. 


124. Graphical representation of scientific data. The 
method of this chapter for representing a mathematical function 
has been adopted by nearly all scientific men to show simul- 
taneous changes in related quantities. Thus, physicists, 
chemists, engineers, statisticians, economists, historians, and 
others, use graphs to present to the eye relations that could 
not be shown otherwise without considerable effort. 


PROBLEMS 
APPLICATIONS IN LIFE INSURANCE 


1. The number of persons per hundred thousand living at 
age 10 that reach certain assigned ages, as given by the Ameri- 
can Experience Table of Mortality, is shown in the following 
table to the nearest thousand : 


Ages | 10} 15 | 20] 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80 | 85 | 90 
No. of thousands living| 100 | 96 | 93 | 89 | 85 | 82 | 78 | 74 | 70 | 65 | 58 | 49 | 39 | 26 | 14| 6: fSk 





Exhibit the relation between the number living and age. 


SuagEstion: Let 20 years be represented by 1 inch along the horizon- 
tal, and 20,000 persons living by 1 inch along the vertical. 


190 RELATION BETWEEN TWO VARIABLES [Cuap. XIX. 


2. A man aged 20 may have his life insured in a certain 
company by the payment of $18.50 per year for $1000 of in- 
surance. The following table gives the amounts that must be 
paid if the insurance is taken at other ages: 


Age. Payment. Age. Payment. - 
20 $20.10 A5 $37.80 
30 22.92 50 46.65 
30 26.55 55 58.84 
40 31.38 60 75.77 


Construct the graph to show the relation between the payments 
and the age of the person when insurance is taken. 


APPLICATIONS TO HISTORICAL STATISTICS 


3. The population of the United States as found from the 
various Censuses is given by the following table : 


Year |1790| 1800] 1810/1820] 1830|1840|1850|1860|1870|1880|1890|1900|1910 
Population in millions | 3 | 4.3 | 7.2 | 9.6 [12.9 |17.1 |23.2 |31.4 [38.6 150.2 |62.6 |76.3 |92.0 








Represent the population graphically. 


4. The marriage rate in England per thousand population 
for years from 1872-1890 is given by the following : 


Year | 1872 | 1873 | 1874 | 1875 | 1876 | 1877 | 1878 | 1879 | 1880 | 1881 
Rate | 17.4 | 17.6 [17.0 [16.7 | 16.5 | 15.7 [15.2 |144 [149 | 154 


Year__| 1882 | 1883 | 1884 | 1885 | 1886 | 1887 | 1888 | 1889 | 1890 
Rate [15.5 [15.5 [15.1 [14.5 [142 [14.4 [144 [15.0 | 155 








Represent this table by a graph. 


5. The following table gives the production of coal in the 
United States in millions of tons for various dates : 


Date | 1850 | 1855 | 1860 | 1865 | 1870 | 1875 | 1880 | 1885 | 1890 | 1895 | 1900 | 1905 | 1910 
Tons of coal| 6.3 | 11.5] 13. | 21.2 | 29.5 | 46.7 | 63.8 | 99.3 [140.9 |172.4 |240,.8|350.6 [447.6 





Construct a graph to show changes in coal production from 
1850-1910. 


es 





ie 


Be Apr. 124] APPLICATIONS 191 


6. In considering the different items that enter into the 
high cost of living, a man records the rents per month he has 
paid for a house during the past twenty years as follows : 


1893 | 1896 | 1900 | 1903 | 1905 | 1909 | 1913 
$15 | $20 | $24 | $28 | $30 | $40 | $55 


Construct a graph to show increase in rent. 


APPLICATIONS TO WEATHER REPORTS 


7. The following table gives the temperatures on a Fahren- 
heit thermometer at a certain station at various hours of the 
day : 
emer) | 3 |4|/5|6|7{8| 9 | 10) 11| 12 |lam] 2 [48 

10° | 11° |10.5°|9.5°|8°|6°}4°|2°|0°| — 3°|— 6° — 9°|— 12.5°|— 15° |-18°|— 20° 
Meise) 6 | 7 | § | 9 | 10 |-11 [Noon 





ee 22 91 .5°[— 21°|— 20°|— 18°|— 15°|— 12°| — 89 


_ Construct the graph to show the changes in temperature with 


respect tothe time. Sucha graph is called a temperature curve. 
By reference to the graph, give approximately the lowest and 


highest temperatures of the day. At about what time was 


the temperature lowest? 
8. Observe the weather report in a daily paper, and draw 
the temperature curve for the 24 hours covered by the report. 


CHAPTER XX 


SYSTEMS OF LINEAR EQUATIONS 


125. Solution of equations in two or more unknowns. By 
a solution of an equation in two or more unknowns, we mean 
a set of values of the unknowns that satisfies the equation. 


Thus, fs % i is a solution of x + y = 5. 


A single equation in two unknowns has many solutions. 


Thus, the equation x+y=5 
: x =0 Z=2 x=3 x=A4 
has solutions, E e 5) (* 3 ai & z al C i iy and so on. 


In fact, an equation such as x + y = 5 in two unknowns 
has an unlimited number of solutions, since if we assign any 
value to x, we can find a corresponding value of y that satisfies 
the equation. Graphically, we may say the coordinates of 
each of the unlimited number of points on the line in Fig. 26 
satisfy the equation 2 + y = 5. 


EXERCISES 


Find four solutions of each of the following equations : 


1, 2 + /2y = 9. 6. s+i = 12. 11. 27 + dy = 5. 
2. y+ 3x = 8. 7 2r+s=7. 12. 8& —y = 6. 
3. 4% + 3y = 5. 8. 27 — 3y = 10. 13. 54 +49 = 
4. 5 + 2n = 11. 9708 ob ais. 14. 6% — y = 5. 
5 v—we=i7. 10. /+4m=12. 15. 1+m=10, 


192 


« 


iJ 


fe Ants. 126, 127, 128, 129] SIMULTANEOUS EQUATIONS 193 




















Fig. 26 


126. Simultaneous equations. In the last article, we have 
shown that an equation in two unknowns has an unlimited 


number of solutions. Two such equations, say x + y = 5 and 


2y —x = 4, are said to be simultaneous equations when at 


least one pair of values of x and y satisfies both. 


The graphs of these two equations are shown in Fig. 27, and 
we have found graphically the solutions of some simultaneous 
linear equations in Art. 123. In the present chapter, such 
equations will be solved by algebraic methods. 

127. Independent equations. Two equations are said to 
be independent if they have distinct graphs. Thus, the equa- 
tions x + y = 5 and 2y — x = 4 (Fig. 27) are independent. 

128. Dependent or equivalent equations. Two equations 
are dependent.or equivalent if they have the same graph. Thus, 
the equations x + y = 5 and 2x + 2y = 10 are equivalent. 

129. Inconsistent equations. Two equations such as 
x+y = 5and 2x + 2y = 18 are said to be inconsistent because 


194. SYSTEMS OF LINEAR EQUATIONS [Caap. XX. ~ 











Y’ One space = one unit 


Hig-v2¢ 


there are no values of x and y that satisfy both equations. The 
graphs of « + y = 5 and 2a + 2y = 18 are parallel lines. (Fig. 
28.) | 

By a system of linear equations, we mean a set of two or — 
more linear equations that are to be treated together. A set 
of values of the unknowns that satisfies each equation of a 
system is said to be a solution of the system. 


130. Elimination. Combining equations of a system in y 
such a manner as to get rid of one of the unknowns is called 
elimination. , 

Example. Find two numbers such that 2 times the first plus 3 times 
the second equals 12, and 4 times the first minus 3 times the second 
equals 6. 

SoLuTion: Let x = the first number, 
and y = the second number. 

Then 2x + 3y = 12, (1) 
4x — 3y = 6. (2) 
Adding, 6x = 18, 
2 eo. | 


a 


: 
| 


Arts. 130, 131] ELIMINATION 195 
BGbetitutine 3 for x in (1) gives . 
6 + 3y = 12. 
Then 3y = 6, 
and OD 
' CHECK: 2°35 +3°2'4 12, 
4-3 —-3:2 =6. 


ae 3 and 2 are the numbers sought. 


















































131. Elimination by addition and subtraction. The exam- 
ple of Art. 130 is there solved by a method of elimination known. 
as elimination by addition. 

To apply the method to a somewhat more general system, 
solve 

ox + 2y = 17; (1) 
4x + 38y = 24. (2) 


These equations are marked (1) and (2). We shall find it 
convenient to write, for the sake of brevity, such statements 








196 SYSTEMS OF LINEAR EQUATIONS [Cuapr. XX. > 


as (1) + 5 to mean the members of (1) divided by 5, and (1) - 5 
to mean the members of (1) multiplied by 5. More generally, 
the symbol (n) is used to denote the members of equation 
marked (n). 


(1) - 3 gives 9x + 6y = 51, (See Art. 38) (3) 
(2) - 2 gives 8x + by = 48, (4) 
(3) — (4) gives GS (5) 
Substituting in (1), 3-3 + 2y = 17, (6) | 
2y = 8, (7) 
¥en (8) 4 


CueEcxk: Substitute x = 3, y= 4 in (1) and (2). This gives 
3°3+2:°4=17, 
4°34+3°4 = 24, 
Hence x = 3, y = 4 is the solution sought. 


Explanation. In elimination by addition or subtraction we 

(1) multiply or divide the members of the equations by such 
numbers as will make the coefficients of the unknowns to be 
eliminated numerically equal. 

(2) We then eliminate by addition if the resulting coefficients 
have unlike signs and by subtraction if they have like signs. 


EXERCISES 
Solve the following systems of equations, making elimina- 
tions by addition or subtraction: 


A. 37 + 4y = 10, SK 42 + 5y = 12, 
t= 62 —y +16 =0. 
\Z. 5a + 3y = 8, 6. 3a 4+ 7b = 7, 
4e + 5y = -1. 5a + 3b = 29. 
3. 62 — 8y = 20, 7 5y—2=242, 
5x + 2y = 8. 2y — 2x +4 =0. 
4. x + 3y = dy — 7, 8. ua =a ; 
x + 2y = 13. 3u — 2v = 10. 





fe 

j 

i 

i 

I 
ea 
j 


|, Arts. 131, 132] ELIMINATION 197 


Seer os. = 1, 12. 15k + 201 = 10, 
4r — 12s = 4. 25k + 141 = 11. 
10. 6p — 5q = 9, 13. 6c + 15d + 6 = 0, 
7p + 2q = 34. 14d + c = 22. 
init —i on = 20, 14. 5s + 6t = 17, 
18m + 6n = 18. 6s + St = 16. 


15. If the coefficients of the letter to be eliminated from 

_ two simultaneous equations are prime to each other, what is 

+ the simplest multiplier for each equation? Answer the same 
question if the coefficients are not prime to each other. 


16. 3x + 4y = 10, 18. 18u + 10v = 60, 
ox + by = 16. 12u — 15v = 105. 
17. lox + 14y = 18, 19. 4m — 15n = 12, 
25% — 2ly = 163. 9m — 10n = 122. 
20. 18u + 10v = 55, 
12u — 15v = 15. 


132. Elimination by substitution. This method of solving 
a system of linear equations is illustrated in the following. 
Example: Solve the system of equations, 


82 —4y =14, © | (1) 
ox + 2y = 32. (2) 
SoLuTion: From (1), 3a = 4y + 14. (3) 
4 14 
From (38), L= te . (4) 
_ Substituting wae for x in (2) gives 
Bey +) + 2y = 32. (5) 
_.(5)* 3 gives 5(4y + 14) + 6y = 96. (6) 
Multiplying, 20y + 70 + 6y = 96. (7) 
_ Collecting, 26y = 26. (8) 
y =1. (9) 
Substituting y = 1 in (4) gives 
2 =6 (10) 


CueEck: Substituting x = 6, y = 1 in (1) and (2), we find the equations 


are satisfied. 


198 SYSTEMS OF LINEAR EQUATIONS [Cuar. XX. 


Solve the following systems by the method of substitution 
and check the results by substituting in the original system of 
equations : 7 








1. + + 2y = 11, 8. $p + 3q = 11, 
or — dy = 3. q-—p= 
oR, 9. eo ny —4, 
ox + 4y = 7. j Sap ae 
3. 5h + 2k = 0, 10. mi + 5m: = 19, 
8h +k = 3. 6m, — 7m, = 3. 
4. Tr — 8s = 5, 11. 27 + 4y = 14, 
r+ l1s= 25. 3a — $y = 24. 
t ae 
5. 8&4 +y =7, fa oa 5 ae 
llz + 2y = 9. it. ooo 
3 aed 
6. 127 + dy = 14, 13. .037 + .70y = 10, 
3x — 10y = 26. O72 + Lely Sie 
Lo aoe 
gh ea i. 2 a 
15a — 5y = 20. Ll? ee 
—~—~+—-=2: 
t.4 
Hint: Solve first fope and ©. 
Ciah amen) : 
tO es eee! 17, (1? ee 
PEATE at LP Y = 
ieee eee 19, 2 See 
ites 0 vt Y 
2 ie Malt o-3 
yaaa ct ae 


+ Arts. 132, 133] GENERAL LINEAR EQUATION 139 


19. 67 + 3y = 19, 21. 4x — Sy = 8, 
8a — 2y = 15. 62 + 3y = 12. 
20. 32 — 7y = 15, 22. 7x + 9y = 2, 
ba + 4y = 11. ox + OY = 2. 

= 49, 

=a 17; 


133. Standard form ax + by +c¢ =0. While some of the 


equations given in the following list of exercises may appear 
complicated, they can all be made to depend on the solution 
of a standard form, 

ax + by +c = 0, 


‘in which all the terms containing x, those containing y, and 


those that contain neither unknown, are collected. When the 
following equations are not given in this form, they should 
usually be reduced to that form as a first step in the solution. 


EXERCISES — 


After reducing the following equations to standard form, 
solve the systems by either process of elimination : 









































_1-4y cy  c+y 16 
5 ple PS . ’ 5. e 3 apie 
3x + 4y = 15. x— 2y = 1. 
7a —15 | 7p+5 Tq-1 
2. 3 = ¥, 6. Bur ats = —9, 
15a — 9y = 27. 10p — g = 2. 
x-3 yt8 
3, 2 - -2, (s m + 5 vy 
Ly eet 2y—4 \ 
| z+ 10y = 5. 9 7 =), 
me, — 8. 5a + Ty = 37, 
= a 6 —2y =8 
2x — 2y = 10. ae oraet lena 


200 SYSTEMS OF LINEAR EQUATIONS [Cuap. xx. 












































9 2x — 1 Cp aS 13. .3p — 2g = 23, 

Oh Se es ee : 2p Ai | ae 
5a — Ay = 12: 3 Gate 
Doe eee 2 + 4 eee 

10. 6 — iF = (), 14. B} 4 = 34, 
i ee ee ot + 45, 27d 
Dee eS is.) ee 

11. 5x + 9y = 40, da +1 40 ae 
Deaths). 15. 5 ae 3 = 153, 
3 ek 4a — 2b =2 

12. 16x — 4y = 150, 16, he 
2x —1 AG er 3 3 4 
ab ah air Qs + 5r = 45. 

PROBLEMS 


1. Find two numbers whose sum is 100 and whose differ- 
ence is 28. 

2. In a meeting of 484 voters a motion was carried by a 
majority of 32. How many voted aye, and how many no? 

3. A farmer paid 3 men and 2 boys $8 for a day’s work, 
and afterwards paid 5 boys and 2 men $9 for a day’s work. 
What were the wages of a man and what the wages of a boy? 

4. ‘The admission to a circus is 50¢ for adults and 25¢ for 
children. If the proceeds from the sale of 5000 tickets is $2200, ; 
how many tickets of each kind were sold? | 

5. The perimeter of a rectangle is 200 feet and the length 
is 28 feet more than the width. Find length and width of the 
rectangle. 

6. A part of $2500 is invested at 6% and the remainder 
at 5%. The yearly income from both is $141. Find the 
amount in each investment. 

7. A part of $5000 is invested at 5% and the remainder at 
4%. 'The income from the part at 4% exceeds that from the 


|, Arr. 133] PROBLEMS 201 


part at 5% by $52. Find the number of dollars in each invest- 
ment. 

8. If A should give B $10, then B would have twice as 
muchas A. If Bshould give A $10, then A and B would have 
the same amounts. How much money has each? 

9. In a certain family each son has as many brothers as 
sisters, but each daughter has twice as many brothers as sisters. 
How many children are in the family? 

10. A bottle and its contents cost 60 cents, and the contents 
cost 44 cents more than the bottle ; what was the cost of the 
bottle? 

11. A mechanic and an apprentice together receive $50 for a 
piece of work. The mechanic works 8 days and the apprentice. 

12 days; and the mechanic earns in 5 days $2 more than the 
entire amount received by the apprentice. What wages per 
day does each receive? 

12. A mule and a donkey were going to market laden with 
wheat. The mule said: “If you give me one measure, I should 
carry twice as much as you ; but if I give you one, we should 
have equal burdens.”’ Tell me what were their burdens. 


e Tradition says that Euclid gave this problem in his lectures at Alexan- 
_ dria 280 B.C. 


13. A and B can do a piece of work in 2 days ; but an equal 
_ piece of work when A puts in only half his time and B only 
one-third his time requires 43 days. How long would the work 
take A and B each, working alone? 
14. A farmer bought 100 acres of land for $4000. If part 
_ of it cost him $34 an acre and the remainder $49 an acre, find 
_ the number of acres bought at each price. 
; 15. A plumber and his helper receive $4.80. The plumber 
i works 5 hours and the helper 6 hours. Working at the same 
- rate per hour at another time the plumber works 8 hours and 
the helper 95 hours, and they receive $7.65. What are the 
wages of each per hour? 


202 SYSTEMS OF LINEAR EQUATIONS [Cuar. XX. . 


PROBLEMS INVOLVING THE LEVER 





Fia. 29 


If two weights w and W balance when placed on a bar at distances 
d and D respectively from the point of support F (called the fulcrum), 
then - 

w:d=W-D 
16. Two weights balance when one is 5 feet and the other 
8 feet from the fulcrum. If the first weight, increased by 25 - 
pounds, be placed 4 feet from the fulcrum, the balance is 
maintained. Find the two weights. 

17. Two children weighing 35 and 49 pounds just balance 
a seesaw board 12 feet long. Where is the support of the 
board placed? 

18. T'wo children, playing on a seesaw board 15 feet long, 
just balance when the support is 9 feet from one end. If the 
child on the long end of the board weighs 50 pounds, what is 
the weight of the other child? 

19. ‘Two unknown weights balance when placed 8 and 10 
feet from the fulcrum of a lever. If their positions are reversed, 
5 pounds 10 ounces must be added to the lesser weight to 
restore the balance. What are the weights? 


PROBLEMS ABOUT DIGITS 


20. In an integer of two digits let ¢ represent the tens’ digit 
and wu the units’ digit. The number is then 10¢+ 4. (a) What 
are t and wu in 46? (6) In 85? (ce) Write each number in the — 
form 10¢+ wu. (d) What is 10¢ + wif t =6 and u = 3? (e) If 
t=landu=97 


, Arr. 133] PROBLEMS 203 


21. In an integer of three digits let h, t, and wu be the hun- 
dreds’, tens’, and units’ digits respectively. (a) Write the num- 
bers. (6) What are h,¢and win 865? (c) In 506? (d) In 189? 
(e) Write each of these numbers in the form 100h + 10¢ + wu. 
(f) What is 100h + 10¢+uifh=9,t=4,u=3? (g) Ifh=1, 
b= 8, u = 0? 

22. A number contains two digits. The units’ digit is 3 
greater than the tens’ digit. ‘The number equals 4 times the 
sum of the digits. Find the number. 

23. The sum of the digits in a certain two-digit number is 
8. If 18 be added to the number, the result is expressed by 
the digits in the reverse order. Find the number. 

24. Two numbers are written with the same two digits; the 


difference of the two numbers is 45 and the sum of the digits 


is 9. What are the numbers? 

25. The numerator and denominator of a certain proper 
fraction each consists of the same two figures whose sum is 9, 
written in different orders. If the value of the fraction is 4, 
find the numerator and denominator. 


PROBLEMS ABOUT COINS 


26. A man has 22 coins amounting to $10, all dollars and 
quarters. How many of each denomination has he? 

27. A collection of nickels and dimes, containing 121 coins, 
amounts to $7.90. How many coins of each kind are there? 

28. A man met some tramps and wished to give them a 
quarter each, but found he had 23 cents too little for that. He 


_ therefore gave them two dimes each and had 42 cents over. 


How much money had he and how many tramps were there? 


PROBLEMS INVOLVING MOTION 
29. Two boys run a race of 440 yards. In the first 


trial A gives B a start of 65 yards and wins by 20 seconds. In 


the second trial A gives B a start of 34 seconds and B wins by 
8 yards. Find the rates of A and B in yards per second. 


204 SYSTEMS OF LINEAR EQUATIONS [Cuap. Xx. 


30. Inamile race A gives B a start of 44 yards and is beaten 
by 1 second. In a second trial A gives B a start of 6 seconds, 
and beats him by 9% yards. Find the number of yards each 
runs in a second. 

31. A train traveling 30 miles an hour takes 21 minutes 


a 


longer to go from A to B than a train which travels 36 miles — 


an hour. Find the distance from A to B. 

32. A steamer makes 50 miles downstream in two hours, 
and returns in 23 hours. Find the rate of the current and the 
rate of the steamer in still water. 

Hint: Let x = rate of steamer in still water in miles an hour, and y 


the rate of the current. Then the rate downstream is x + y and the rate 
upstream is x — y. 


33. A boat goes downstream 72 miles in 3 hours, and up- 
stream 48 miles in 3 hours. Find the rate in still water and 
the rate of the current. 


PROBLEMS ON MIXTURES 


34. A grocer has two kinds of sugar, one worth 5¢ and the 
other 6¢ a pound. How many pounds of each sort must be 
taken to make a mixture of 25 pounds worth $1.40? 

35. What quantities of silver 72% pure and 84.8% pure 
must be mixed to give 8 ounces of silver 80% pure? 

36. If 25 pounds of sugar and 10 pounds of coffee together 
cost $5.50, and at the same price 25 pounds of coffee and 10 
pounds of sugar cost $10.60, what is the price of each per pound? 

37. A farmer finds that one day with 200 pounds of milk 
and 40 pounds of cream, he gets 24 pounds of butter. On 


ee) 


another day with 150 pounds of milk and 25 pounds of cream, i 
he gets 16 pounds of butter. (a) What per cent of his milk is © 


butter? (6) What per cent of his cream is butter? 


38. How many gallons each of cream 40% fat and milk | 
5% fat shall be mixed to produce 30 gallons a the mixture — 


162% fat? 


| 


Arts. 133, 134] LITERAL EQUATIONS 205 


39. A pound of tea and 25 pounds of sugar cost $1.75. If 
sugar rises in price 20 per cent and tea 10 per cent, the same 
amounts cost $2.05. Find the price per pound of each. 

134. Literal equations containing two unknowns. When 
the coefficients of the unknowns are letters, the equations with 
two unknowns can still be solved by means of elimination, but 
the coefficients of the unknowns appear in the results. 


EXERCISES AND PROBLEMS 


In the following exercises, consider the letters a, b, c, d 
from the first of the alphabet as known numbers ; solve for 
the x, y, 2, w and check: 














1. © + 3y = 2a, (1) 
10x — y = 3a. (2) 
SoLuTION: (2) - 3 gives 30x — 3y = 9a. (3) 
(1) + (8) gives slz = lla, (4) 
~ ‘Lin 
aay be (5) 
re lla 
Substituting in (1), 31 + 3y = 2a, (6) 
ae an, 
_ la si 
egy 
, lla Iva _ 1ila+5la_ 62a_ 
CHECK: 31 +3 Thos 31 = The 2a. 
tig, 17a 110a—17a 93a 
Seviaoesl so 8i, kr at Ot 
2. 4x — 2y = 5b, 4. 3.5” + 3y = 5a, 
6x+3y 7b. x+y = 3a. 
3. ax + 3y = 10a, 5. ox — 4y = lla, 
4x — 2y = 6a. x + 3y = 6a. 
6. 8x + Sy = 4a + 96, 
x a — 126 
ey rnd) 





2 a 


206 SYSTEMS OF LINEAR EQUATIONS [Caap. XX. 


7 se 0 eee 9. ax + by =<, 

a Cpe . dx + ey =f. 
bz fw 38. ee 

Cie om 10. —+-— = 2a, 
x» 

8. ax + by = 0, 2 3 _ 3p 
xty+tc= bc 


11. The base of a triangle is 10 and the altitude is 8. Find 
the area. 





12. The base of a triangle is a and the altitude is h, what is , 


the area? 

13. The base of a triangle is a and the altitude is 10. What 
is the altitude of a triangle of the same area but with a base 
a+ 2? 

14. The altitude of a triangle is h and the base is a. If 
the base be increased by b, how much must the altitude be 
decreased so as to leave the area unchanged? 

15. The sum of two numbers is s; their difference is d. 
Find the two numbers. 

16. Two persons, A and B, can complete a certain amount 
of work in 8 days; they work together 4 days; B finishes it 
in 5 days. Find the time each would require to do it alone. 

17. Two persons, A and B, can complete a certain amount 
of work in 1 days; they work together m days, when A stops ; 
B finishes it in n days. Find the time each would require to 
do it alone. 


— | 


18. Determine 6 and c so that 2? + bx + ¢ is equal to 


1 when x = 1 and is equal to 2 when z = 2. 


19. Two books cost a dollars. The one cost b dollars more 7 


than the other. Find the cost of each. 


20. If A gives d dollars to B, they have equal sums. If | 
B gives e dollars to A, then A has 3 times as much as B. How — 


much has each? 


135. Linear systems in three or more unknowns. ‘To solve 


a system of three equations, involving three unknowns, one of 


ae Ss 


|. Arr. 135] EQUATIONS IN THREE UNKNOWNS 207 


ine 


the unknowns must be eliminated between two pairs of the equa- 
tions. The problem is then reduced to one of two unknowns. 
Likewise, to solve a system of four equations, involving four 
unknowns, one unknown must be selected for elimination, 
and it must be eliminated from three pairs of the equations. 
We have then three equations with three unknowns and proceed 
as above to reduce them to two equations with two unknowns. 


Example : | 
Solve 22 + 4y + 32 = 8, (1) 
x—-d5y+z2 = -4, (2) 
3x2 — 10y + 5z = -3. (3) 
Soturion: Eliminate one unknown, say z, between (1) and (2), thus: 
2) > 3, 3a — 1l5y + 32 = —12, (4) 
(1), 2x + 4y + 32 = 8, (5) 
(4) — (5), x —19y = —20. (6) 
Now eliminate z from (2) and (3) as follows: 
(2).- 5, 5a — 25y + 52 = —20, (7) 
(3), 3a — 10y + 52 = -3, (8) 
(7) — (8), 2a — ld5y = —-17. (9) 
The equations (6) and (9) contain only the unknowns 2 and y. 
(6) ~ 2, 27 — 38y = —40, (10) 
(9), 27 — 15y = —17, (11) 
(10) -— (11), — 23y = —23, (12) 
(12) + —23, y =1. (13) 
_ Substituting 1 for y in (9), 
27 —15 = -17, - (14) 
Solving (14), z= -l. (15) 
Substituting z = -1, y =1 in (1), we get 
a+ 4 4 oe = 8, (16) 
or 9) Ba'= 6. (17) 
(17) +3, 2 = 2. ; (18) 


Cueck: -2+4+6=8, or 8 =8; 
-1-54+2=-4, or —4 = -4; 
Seeetio +10 = —3, or | -3.= -3. 


208 SYSTEMS OF LINEAR EQUATIONS [Cuap. XX. 


By 
EXERCISES AND PROBLEMS 
1. ty ee ee 5 1, 2= ee 
3x + 3y — 2z = 60, Te 2 4 
10x — 5y — 32 = 0. 5 8 See 
2. 8x2 — Sy — 22 = 14, 2 
ox — 8y — z = 12, 7 4 8 
c— 3y — 32% 1. ier 


3. ¢+y+2=6, ae 
27 —-y—2= -8, Hint: Solve first for ie 


x + 2y + 32 = 14. 


4. 4x —2y +2 = —6, 6 r+yt+z=4a, 
xt+y+z=0, 3x — l2y — z = 2a, 
2x — 3y + 32 = 2. ox + 3y —zZ=4. 

7 crc+y+e2e+uw=d0O0, 8 r+yt+2+w =, 
x+2y—z2+3w =0, x + 2y + 32 + 4w = 17, 
2x +y —-22+w =4. 4x + 3y + 22+ w =8, 
2x + 4y —2+ 3w = 5, x — 2y+ 32 — 2w =83. 


9. Find three numbers such that the sums formed by tak- 
ing them in pairs are 30, 40 and 50. 

10. The sum of three numbers is 76. The sum of the first 
and second is 4 greater than the third number, and the differ- 
ence of the first and second is one-third of the third number. } 
Find the numbers. 

11. The perimeter of a triangle is 74. The sum of two — 
sides is greater by 10 than the third side, and the difference of — 
the same two sides is 10 less than the third side. Find the 
sides of the triangle. = 

12. Divide 1000 into three parts, such that the sum of the 
first, 3 of the second, and /5 of the third shall be 400; and the 
sum of the second, § of the first, and 75 of the third shall be 450. — 

13. Three cities, connected by straight roads, are at the — 
vertices of a triangle. From A to B by way of C is 112 miles; _ 







~ Arr. 135] EXERCISES AND PROBLEMS 209 


from B to C by way of A is 116 miles ; from C to A by way of B 
is 104 miles. How far apart are the cities? 

14. Separate 400 into 4 parts such that if the first part be 
increased by 9, the second diminished by 9, the third multiplied 
by 9, and the fourth divided by 9, the results will all be equal. 

15. In a race of 500 yards, A can beat B by 20 yards, and 
C by 30 yards. By how many yards can B beat C? 

16. Between two towns the road is level one-half of the dis- 
_ tance, and the speeds of a motor car are 9, 20 and 18 miles per 
hour up hill, on the level, and down hill. It takes 54 hours to 
go and 6% to return. What are the lengths of the level and 
inclined parts of the road? 


MISCELLANEOUS EXERCISES AND PROBLEMS 


Solve the following equations for x and y when the solution 
is possible. Show by means of a graph why the solution is 
impossible in Exercises 4, 6, 8, and 15. 











1. 6x — 3y = 15, 6. Se nN Mea 
2x + 7y = 45. 6 5 
r+d yt+6 
x-1_ (y-3) Dpewiwgagiay oF 
ae aaa 5) = 4) 
 2— y= 2, 
mec tt > 2x — 2y = 10. 
B ul 8. y + 22 = 7, 
ame — 9 = 23, 2y + 4x = 4. 
az + 5y = —13. 9. ax+y=a+2, 
x+ay = 2a +1. 
ome Ya 10. ax + y = 2a, 
3x + dy = 6 2ax —y =4.. 
De fe 11. 3x — 4y = 2a, 
Beata * 4e + 3y = 11a. 
foes -'5 12. ax — by = 0, 
Pes... bx — ay = b? — a’. 


210 SYSTEMS OF LINEAR EQUATIONS [Cuap. XX. * 











13. ax + y =, 15. 2 + 3y = 2, 
bx +y =a. 32 + 9y = 15.. 
dame pesd Pas ot 8 2 oo 

1d. = ep ee ey 
2 nly eieh si 0) 
db 4 ee ye 


17. Find two numbers whose sum is 60, and such that one 
of them exceeds twice the other by 6. 

18. A board 18 feet long is cut into two pieces whose lengths 
are in the ratio 1 to 3. How long are the pieces? 

19. A banker changes $5.00 into dimes and nickels. ‘There ~ 
are 73 coins in all. How many dimes and how many nickels 
are there? 

20. There are two numbers whose sum is 63. If the greater 
is divided by the smaller, the quotient is 2 and the remainder 
9. What are the numbers? | 

21. A board a feet long is cut into two pieces whose lengths 
are in the ratio b to c. How long are the pieces in terms of a, 

b, and c? 

22. A grocer bought oranges, some at 20 cents a dozen and 
some at 18 centsa dozen. He paid for all $6.70. He sold them 
at 25 cents a dozen and cleared $2.05. How many oranges / 
did he buy at each price? | 

23. After an examination a teacher decided to raise each 
grade from x to y by the formula y = mz + b, where m and 6 
are to be determined by the facts that a boy who made 50 is 
to receive 65 and one who made 60 is to receive 77. Find m 
and b, also the new grade of a boy who received 75. . 

24. A bird flying with the wind makes 60 miles an hour, 
but when flying against a wind half as strong it makes only _ 
45 miles an hour. Find rates of the two winds. Also the rate _ 
of the bird in still air. | 

AssuMPTION: It is to be assumed that the rate of the wind should be 


added to the rate in still air when the bird goes with the wind and sub- % 
tracted when it goes against the wind. if 


Reisen ee 
ie ee 


_ Art. 135] PROBLEMS 211 


25. The report from a pistol travels 1080 feet per second 
with the wind, and 1040 against the same wind. Find the 
rate of the wind and the rate of sound in still air. 

26. An aéroplane flies with the wind at the rate of 80 miles 
per hour and against a wind twice as strong at the rate of 50 
miles per hour. Find its rate in still air. 

27. A certain sum of money is invested at 5 per cent and 
another at 6 per cent. The annual income from both invest- 
ments is $98. If the first sum had been invested at 6 per cent 
and the second at 5 per cent, the income would have been $2 
greater. What are the two sums of money? 

28. The grocer sold Mrs. Brown 3 quarts of strawberries 
and 2 quarts of cherries for $0.65. He sold Mrs. Jones 2 quarts 
of strawberries and 5 quarts of cherries for $0.80. Find the 
price of a quart of strawberries and of a quart of cherries. 

29. The grocer sold Mrs. Brown 8 quarts of strawberries 
and 2 quarts of cherries for a certain sum of money. He sold 
Mrs. Jones 1 quart of strawberries and 5 quarts of cherries for 
the same sum of money. Compare the price of a quart of straw- 
berries and the price of a quart of cherries. 


212 SYSTEMS OF LINEAR EQUATIONS [C#ap. XX. 


REVIEW EXERCISES AND PROBLEMS 


1. Find amean proportional to each of the following pairs of numbers : 
—3, -12; 8,4; mr, wR; 12%, a? ; 52%, 20y?. 


2. Solveforz: ax =b;—=b; a+x=b;2x-a=b. Tellin each 


case which of the principles 4 Art. 38 is used in solving the equation. 


3. Give an example of a linear equation in two unknowns. What 
is meant by a system of linear equations? Give an example. Define a 
solution of an equation in two unknowns. How many solutions can be 
found for 2x — y = 10? When are two linear equations in two unknowns 
said to be simultaneous? Give an example. When inconsistent? Give 
an example. 


4. What is the locus of an equation of the form az + by =c? How 
many points must be found to determine the locus? Are the a 
pairs of equations simultaneous? Do their loci intersect? 


x+y=4 4 b= 2 z=1 
@ {e+ seks Oo ike @{% ag © {9 a 
5. What changes of sign can be made in the erases ; Without chang- 


wr 3 A a 
ing its value? Answer the same question for = 


6. Draw a pair of codrdinate axes, XX’ and YY’, which intersect at 
O and divide the paper into four parts. Show the part in which a point 
is located if (a) its codrdinates are both positive ; (b) both negative ; (c) 
the abscissa is positive and the ordinate negative ; (d) the abscissa is nega- 
tive and the ordinate positive; (e) both zero; (f) the abscissa is zero ; 
(g) the ordinate is zero. 


7. State an equation giving the relation between # and y if (a) y 
is twice as great as x; (b) yisk times as great as x; (c)if y varies as 2 


(d) What does the last relation become if y= 16 when x = 8? 


8. If y varies as the les of xz, and y = 8 when zw = 2, find y when 


x has the values 0, .01, .1, 4, 2, 1, 2, 3, 10. 


9. Make a statement concerning the proportionality of area and 
one concerning the proportionality of areas of similar triangles. Tlus- 


trate by figures the meaning of each statement. 


> 





#3 


~ 


10. The hypotenuse of a right triangle is 5 inches and one side is | } 


inches. Find the other side. The hypotenuse of a similar triangle is 10 
inches. Find the other sides. 





| 


: 


ee 


} 
a 
| 
; 


} Art. 135] REVIEW EXERCISES AND PROBLEMS 213 


11. A man 6 feet tall is standing 10 feet from a lamp post which is 15 
_ feet high. Find the length of his shadow. 


12. A tree casts a shadow 60 feet long when a post 10 feet high casts ° 
a shadow 8 feet long. How high is the tree? 


13. Show that in the proportion a:b = c:d, the product of the means 
divided by either extreme equals the other extreme ; and that the product 
of the extremes divided by either mean equals the other mean. 


Solve the following equations : 
14. 2x -— 3x = $4 — 3 -— 40 +2. 
meets — se 7 1 


a'3 82 12 42 
16. ax+br =m+2. 
F 17. (a —2x)(b — 2) = 2’. 
ox —-5 Se-1l ax-4 
een ye 7! 1 
19. 5% + 3y + 2 = 0, 
oz + 2y +1 =0. 
. 20. 242 = 33y + 4, 
2ly = 33a — 47. 
21. +1 =2(y +1), 
y+2=4(2+4+1), 
2+3 =2(% +1). 


22. Determine 6 and c in the equation y = x? + bx +c, if y=1 when 
x = —1,and y = 5 whenz = 1. 














2, 


23. Determine a, b, and c in the equation y = az? + bx +c, if y =1 
when x = 0, y = 3 when x = 1, and y = 6 when gz = 2. 


24. The volume of a cylinder varies as the square of the radius when 
its height is constant. When the radius is 1, the volume is 153. Find 
the volume when the radius is 6. 


CHAPTER XXI 
SQUARE ROOT AND APPLICATIONS 


136. Definition of a square root. A square root of a num- 
ber is one of its two equal factors. 


Thus, 2 is a square root of 4, since 2-2 = 4, and 2a is a square root of 
4a?, since 2a : 2a = 4a’. 

Since a? = (—a)? = —a- —a, it follows that every square 
has two square roots, differing only in sign. 


Thus, —2 and +2 are both square roots of 4. 


137. Radical sign. The radical sign ~/  —_ is _ used to 
indicate the positive square root of the number under it. When 
a negative root is to be taken, the radical sign is preceded by 
the sign —. 


Thus, ++/4 or./4 means 2, and —/4 means —2. 


EXERCISES 
Find the following roots by inspection : 
ih, cy eli} 5. +/49. 9. 81. 
2. —/64. 6. /a°b?. 10. —+/225; 
3, —/al. 7. /100. 11. —+/169.m 
4. 4/121. 8. —4+/144. 12. +4/2R6 


138. Square root of monomials. The square root of a 
product may be found by finding the square root of each of its 
factors, and then taking the product of these roots. The type 
form may be written | 

Vab = Va‘ Vb. q 
214 


A Arts. 138, 139] SQUARE ROOTS 215 


This principle is of much value in finding the square root 
of a monomial, if it consists of factors each of which is a square. 
In fact, we may find the square root by simply dividing the 
exponent of each factor by 2. 





Thus, \/225 =/9-25 = /#8 - & =3-5 =15, 
and / Marys = /9 /B/P = Ba3y'. 
EXERCISES 


Find expressions equal to each of the following and free of 
radicals : 


1. /144. 8. —+/36rty%2®. 15. \/64r oy? 
9. 4/256. 9. /8latt® 16. +/64a%®. 
gee 4/025. 10. 1/490 17. . v/xtysz!0, 
Wee s/ 1225. 11. —V/2y"_ 18. —+/25- 36zty®. 
5. -V/Oaty’. 12, fa 19. /64° 8127! 
6. —+/16a°d®. 13. ~/3%r8y4, 20. —/amb?c!. 
Tee oe at. 14. —V/4exef 21. 22527. 


139. Equations solved by finding square roots. Since +2 
and —2 have the same square, 4, the equation 


, y= 4 


is satisfied by both +2 and —2. 


That is, ih 
£V4 = +2, 


are two solutions of the equation x? = 4. 


EXERCISES 
Solve the following equations : 
to =.20. bg Dike 
goer = 121. 6. a? = 81a‘d§. 


3. 2? — 144 = 0. 6. x? = 42a‘b?. 


216 SQUARE ROOT AND APPLICATIONS [Cuap. XXI. 1 


Tate 12. x? — 64a7b® = 0. 
5.8 a= 0r Us 13. x? = 16(a — b)?. 
9. a? — 225a‘b8 = 0. 14. 2? = 49a%D°, 
10) 7 = 3ba°O: 16.2" = Sige 
11. x? = 100(a + D)?. 16, =a 


140. Square roots of trinomials. If a trinomial is a perfect 
square, its root is easily extracted by comparison with the | 
familiar formula 


a? + 2ab + b? = (a + b)?, 
at b. 


from which a/a? + 2ab + 6b? 


Thus, to find 4/92? + 1l2zy + 4y’, 
we may make the comparison by putting 


/92? = 3x =a, 
and /4y? = 2y = b. 
Since 12ry = 2ab, we have 





/9x? + 12ry + 4y? = 8x + 2y. 


EXERCISES 


Extract the square root of the following by inspection : 


1. 4a? + 12ab + 967. 4. 42° + 4zy + y?. 
2. 16x? + 247 + 9. §. 477+ 47 + 1. 
3. c@ — 4ac + 4a’. 6. 9m? — 6mx + 2. 


141. Process of finding the square root. Given that a + 6 
is a square root of a? + 2ab + b?, it is well to follow a certain 
process by which a + 6 may be obtained from a? + 2ab + 6?. 


PROCESS 
a +2ab +0? | a+b 
a? 
Trial divisor = 2a Zab + Bb? 
Complete divisor = 2a +b 2ab +b? . 





} 
i 
} 


| Arr. 141] SQUARE ROOT OF POLYNOMIALS 217 


To follow the process indicated, let us note that the first 
term, a, of the root may be obtained by taking the square root 
of a certain term, a”, of the given expression. 

If a2 is subtracted from the given expression, the remainder 

is 2ab + b. 
The second term, b, of the root may be found by dividing 
a certain term of the remainder by 2a, which is twice the part 
of the root already found. 
| On this account, twice the root already found is called the 
trial divisor. 
iN Since the remainder 2ab + b? = b(2a +b), the complete 
divisor which multiplied by 6, produces 2ab + 6’, is 2a + 6. 
The complete divisor is thus found by adding the second term 
of the root to the trial divisor. 

Before trying to extract a square root of a polynomial, the 
terms should be arranged according to ascending or descending 
powers of some letter. 

Example 1. Extract the square root of 


162? — 24xy + Sy? 
by following the process just explained. 





| PROCESS 
: 162? — 24ry + 9y* | 4a — dy 
162? 
Trial divisor = 8x —24xy + Oy’ 
Complete divisor = 8x — 3y —24zry + 9y? 


Example 2. Extract the square root of 
1622 — 24ry + Oy? + 16az — 12yz + 42’. (1) 


In squaring 4% — 3y + 2z, we may treat 4x — 3y as a single term, 
and write 
{ (4% — 3y) + Qz}2 = (4x — By)? + 42(4e — 8y) +42’. (2) 
Hence, in extracting the square root of (1), we may find first the square 
root of the first three terms as in Example 1. Then to find the next term, 
2z, it is seen from (2) that we should use 2(4x% — 3y) as a trial divisor. 
That is, twice the part of the root already found should be used as a trial 
divisor. 


218 SQUARE ROOT AND APPLICATIONS [Cuap. XXI. 


PROCESS 
16x? — 24xry + Oy? + 16xz — 12yz + 42? [4a — By 4 22 
162? 
Trial divisor 
= Sr —24ary + 9y? 
Complete divisor 
= 8r — 3y —24ry + 97? 


Trial divisor 
= 8x — by 16xz — 12yz + 42? 
Complete divisor 
= 8x — by + 2z l6rz — 12yz + 42 


Explanation. We first proceed as when the root is a binomial, and 
find, for the first two terms in the root, the expression 4a — 3y. 


\ 


y 


Take twice this root already found as a trial divisor. Hence, the 


trial divisor is 8a — 6y, and the next term of the root is 2z. 

Adding this to the trial divisor, we obtain 8x — 6y + 2z for a complete 
divisor. 

Multiplying this by 2z, and subtracting the result as shown, we have 
no remainder. 

In case the root contains more than three terms, the process indicated 
is continued. 


EXERCISES 
Extract the square roots of the following and verify : 
25a? + 30a + 9. 4. Quy? + 12xyz + 42. 
494 — 282? + 4. 5. 9a*t — 4223 + 4927. 
Ox? + 80ry + 25y?. 6. 25y? — 40y + 16. 
at — 2a?b + 2a®c? — 2bc? + b? + c*. 
9a? + 256? + 9c? — 30ab + 18ac — 30be. 
4y*y? + 12a*y + 92? — 30ry? — 20xy? + 25y4. 
10. 1 + 2% 4+ 7a? + 623 + Oat. 
11. 9a* — 12a3y + 3407y? — 20ry? + 25y4. 
12. x* — 10a? + 212? + 20% + 4. 
ie ete 
2 2216 
14. 4x4 — 1223 — 72? + 242 + 16. 


SO eam Poa: c. 


13; 24) 223.4 








/ Anrs. 141, 142] SQUARE ROOTS 219 


15. 9x? — 302 + 4 ~ i desi 


16. 252? + 30ry? + 9y* — 2.5x2 — 1.5y?z + 0.06252". 


dara? Qax® x4 


asd 30 8 
at 4x3 xy 2 


Y 
ae om 3 gs oo 
18. 9 g +3 +4 ry + 7 


19.. x — 40° + 5a? — 27 + 5. 


| 

s 4 3 “ee 1 
f 20. 92* — 122 +42? —-- + = + 6, 
zZ a 





at 
2 lyf 4 + a®xe.+ 





21. =+% z ue 

22. a® — 6a* + 15a* — 20a? + 15a? — 6a + 1. 

He 98. 1 + Qr + 3a? + 423 + 5at + 405 + 3x8 + 227 + 2. 
24, 1624 — 2473 + 25a? — 122 + 4. 

25. x" + Qarry” + y". 


ih 142. Square roots of numbers expressed in Arabic figures. 
| The positive square root of 100 is 10; of 10,000 is 100; of 
1,000,000 is 1000; and so on. Hence, the square root of a 
number between 1 and 100 is between 1 and 10 ; the square root 
of a number between 100 and 10,000 is between 10 and 100; the 
| square root of a number between 10,000 and 1,000,000 is between 
100 and 1000; andsoon. That is to say, the integral part of the 
: square root of a number of one or two figures, contains one 
figure; of a number of three or four figures, contains two figures; 
of a number of five or six figures, contains three figures ; and 
so on. 
Hence, if an integral number is separated into periods of two 
figures each, beginning at the right, its square root has as many — 
digits as the number has periods. 





220 SQUARE ROOT AND APPLICATIONS [Cuap. XXI. + 


143. Explanation of process of finding square root in arith- 
metic. ‘The process of finding square roots of numbers of 
arithmetic may now be explained by the process just used for 
polynomials. 


_ Example. Find the square root of 576. 


SoLtutTion: Separating into periods of two figures each as indicated in 
Art. 142, we have 
5’76. 


There are thus two figures in the integral part of the root. 
To explain the solution algebraically, we may use the formula : 





(¢+u)? = + 2tu+w = + (2 + u)u, ct 


in which ¢ is the number represented by tens’ digit, and wu is the number 
represented by units’ digit. 
The processes in the columns at the right and left below are alike if 
t = 20 and u = 4. 
O4+2iu+wvilti+u 5’76 | 20 +4 = 24 
e 4 00 


Trial divisor = 2¢ | oe 2t = 40 | 176 
Complete divisor , 
=2i+u|l 2tu+u 2t+u =44| 176 
144. Numbers with more than two periods. The method © 
just explained applies to numbers that separate into more than 
two periods of two figures each. 
In making the application, we consider ¢ in the typical form — 


t* + 2tu + wu? to represent at each step the part of the root already 
found. 


Example. Find the square root of 44944. 
4’49’44 | 200 + 10 + 2 = 212 


4 00 00 ; 
2t = 400 4944 | 
2 +4 = 410 4100 1 
2t = 420 844 


2t +u = 422 844 





‘? Arts. 144, 145] . SQUARE ROOTS OF DECIMALS 221 


Omitting zeros, we may condense the process so as to appear as 





follows: 
; 4'49/44 | 212 
4 
4 49 
41 41 





145. Square roots of decimals. Decimals are separated 
into periods of two figures each from the decimal point towards 
the right; for, if the square root of a number has decimal places, 
the number itself has twice as many decimal places. 


} Example. Find the square root of 5220.0625. 
52’20.’06’25 | 72.25 
49 











When a number is not a perfect square, we may annex 
periods of zeros and continue the process of root extraction. 








EXERCISES 
Find the square root of each of the following : 
1. 3025. 9. 599076. 
2. 508369. 10. 87.4225. 
3. 930.25. - 11. 0.015625. 
4. 96.4324. 12. 0.00321489. 
5. .000625. 13. 75570.01. 
6. 2985984. 14. 20880.25. 
; 7. 65536. 15. 1849. 
! 8. 13107.9601. 16. 250500.25. 


222 SQUARE ROOT AND APPLICATIONS [Cuap. XXI. 


25 144 2809 
17. 10.9561. 19. 36 21. 576. 23. 5184 

225 289 361 
18. 0.001225. 20. 956. Pe 394 24. 1600 


® 


25. To illustrate square root by a diagram, let us think — 


of a square board that contains 529 square inches (Fig. 
30), and try to find the number of inches in a side. What is 


multiple of 10 inches that can be cut 
from the board? When 7’ is found, 
what is the length of one of the rect- 
angles U? How many square inches 
of the 529 are left after T is removed? 
Neglecting U’ as small compared to the 
U’s, how can the width of U be found 
from the area and total length 2 x 20 = 
40 of the 2U’s? What is 2 x 20 called 
in the process of finding the square root? If a side of U’ be 
added to the total length of the two U’s, what is the total 
length of the 2 U’s and U’? What is this number called in the 
process of finding square roots? 

146. Approximate square root. 

Example. [ind the square root of 2, correct to two decimal 
places. 2.’00’00’00 | 1.414 








011900 
2.824] .011296 


Norte. — To get the root correct to two decimal places, it is necessary 


the largest square 7’ whose side is a 





to find the figure in the third decimal place, and to note whether it is : 


larger or smaller than 5. In the present case, it is smaller than 5, and 
1.41 is the approximate value sought. That is to say, we mean by “cor- 


é 


rect to two decimal places,” the same as ‘correct to the nearest hundredth.” — 


4 


4 


Arr. 146] APPLICATIONS 223 


| EXERCISES 

Find square roots of the following, correct to two decimal 
places : 
‘ies 6. 24. tier: 16. 200.02. 

Be D. i 18. Loe, 0.02: L7eeo: 

aa .6. S27. 13. 0.002. 18, 2. 

4, 8. 9. 2.92. 14. 0.005. 19. 2. 

5. 10. 10. 0.305. 15. 0.307. 








SS. ee ee 


20. Show that \/4 +9 = V4+4 V9. * 


APPLICATION TO PROBLEMS FROM MENSURATION 


Nore. Unless otherwise stated, find approximate values in the follow- 
ing problems, correct to two decimal places. 


- Using the letters as shown in Fig. 31, supply missing values 


- of letters in the following : 


gee 0 eS, c= 7 


3. a = 20, b = 300, c=? 
Geo 1, c= 2,b=7 4. a 


a3). = 20), C=! 
5. A boy having lodged his kite in the top of a tree, finds 


that by letting out the whole length of his line, which he knows 
is 225 feet, it will reach the ground 


180 feet from the foot of the tree. 


What is the height of the tree? MK 
6. A tree 80 feet high was 
broken off by a storm, the top 
striking the ground 40 feet from b 
the foot of the tree, and the broken tebe 


end resting on the stump. Assum- Fia. 31 


ing the ground to bea horizontal plane, what is the height of 
the part standing? 
7. Two vessels start from the same point, one sailing due 
northeast at the rate of 15 miles an hour, and the other due 
* The sign ~ is read “ is not equal to.” 


224 SQUARE ROOT AND APPLICATIONS [Cuap. XXTI. 


southeast at the rate of 18 miles an hour. How far are they 

apart at the end of 24 hours, supposing the surface of 
<—_—___—. 60 —_———» _ the earth to be a plane? 

8. Find to thenearest tenth 


of area 1200 square feet. 

9. Find the diagonal (line 
hee BD of Fig. 32) of a rectangle 
B C of sides 60 feet and 30 feet. 
Mee 10. Find the diagonal of a 





square of sides 50 feet. 

11. The diagonal of a square is 24 feet, what is the length 
of its sides? 

12. The dotted line in Fig. 33 indicates a path across a 
field. How many rods aresaved by taking the path instead 
of following the road? 

13. Find approximately (to the nearest tenth of a rod) the 
sides of a square having an area equal to that of a rectangle 
whose sides are 40 rods and 50 rods. 

14. Find the sides of a square having an area equal to that 
of a triangle of base 70 feet 
and altitude 35 feet. 


Facts from Geometry: (1) The 
area of a circle is mr?, where 7 = 
3.1416, and r is the radius of the 
circle. 

(2) The surface of a sphere is 
Amr’, where r is the radius. 

(3) The convex surface of a 


cylinder is 2mrh, where r is the to “a 


radius of base and h the altitude. 


Road 





15. Kind the radius of a circle whose area is (a) 50.2656 — 
square feet; (b) 1000 square feet. 
16. A hase is to be tied to a post in a pasture by means 


of a rope just long enough so that he can graze over 3 acre. 


¢ 


of a foot the sides of a square © 


& 


| 
a. 
7 
» Fi 


» Arr. 146] PROBLEMS 225 


~ How many feet long should the rope be if an allowance of 
' three feet is made for tying? 


(One acre = 48,560 square feet.) 


17. A given circle has a radius of 4 feet. What is the 
radius of a circle whose area is twice that of the given circle? 

18. A bowl 8 inches in diameter in the form of a hemi- 
sphere is made by pressing from a circular piece of brass. What 
is the diameter of the piece of brass required to make the bowl? 

AssumPTION: In Problems 18 and 19, it is assumed that the areas are 
the same before and after pressing. 

19. A shoe-blacking box (without lid) 4 inches in diameter 
and 3 inch deep is pressed from a circular piece of tin. Find 
the diameter of the piece of tin required. 


CHAPTER XXII 
RADICALS 


14%. Radical. An indicated root is called a radical expres- 
sion, or simply a radical. Thus, \/a is a radical expression or 
more briefly a radical. The radical sign is used to indicate 
other roots than square roots by means of a figure called the 
index of the root. Thus, in V27, the number 3 is the index and 
shows that the third or cube root is indicated. 

In general, the nth root of a number a, one of its n equal factors, 
is written ~/a. 

148. Rational and irrational numbers. A rational number 
is an integer or the quotient of two integers. Thus, 2, 2, 3.333 
are rational numbers. 


Exercise. Express 3.333 as the quotient of two integers. 


It is found useful to extend our number system to contain 
numbers that are not rational numbers. For example, if we 
attempt to find the side of a square whose area is 2, we may 
write the result as 1/2, but this is not a rational* number. 

Stated in another form, we have a solution for the equation 


x? = 2 


only when we extend our number system to include more than 


what we have defined as rational numbers. 

Any number which is not a rational number is called an 
irrational number. 

Thus, V/2, V5, /10, ¥/7 are irrational numbers. 


149. Surds. A surd is an irrational number that is a root 


of a rational number. Thus, v/2, /3, \/5 are surds, but »/4 


* See Rietz and Crathorne’s College Algebra, p. 18. 
226 


} 


‘ 
i 
t 
| 
iat 


ny, 
A 


¥ 


Arts. 150, 151] SIMPLIFICATION OF RADICALS 227 


is not a surd since 1/4 = 2, and 1/2 + +/3 is not a surd since 
2+ +3 is not aa rational number. 

Surds that express square roots only are called quadratic 
surds. 


150. Square root of a fraction e Since a fraction is squared 


by squaring the numerator and denominator separately, it 
follows that the square root of a fraction is given by extracting 
the square root of the numerator and denominator separately. 


In symbols, ai 
Vi-% 
6b y/o 


151. Simplification of radicals. The form of a radical 
expression may often be changed to advantage without chang- 
ing its value. 


For example, V8 = V/4-2 = 2/2; 
and if we know that /2 = 1.4144, 
we have /8 = 2(1.414+) = 2.8284. 
Further, CMe 
ee 


and the latter is easier to compute to a given number of decimal places 
than the former. 


To continue with illustrations, let it be required to express \ 3 2PPTOX- 


- imately as a decimal fraction. Three plans suggest themselves, for we 
_ may write 


2 _ v2 V2 = v066064 are a a 
aor REN ea EO NG 1) N 353: ¢ 33 








VJ/2_ 1.4144 | Li 
By the first plan, Oe TRAD S 0.8165 — . 
By the second plan, 1/0.6666+4 = 0.8165 —. 
By the third plan, ve = ae = 0.8165 —. 


The first plan clearly involves more labor in computation than either the 
second or the third plan. 


225) RADICALS (Cuar, XXII 


EXERCISES 


Find, correct to two decimal places, the following square 
roots; do Exercises 1 and 5 by three plans: 


1, 4/3: 4, /8. 2. <p 10. 1/2: 

2. fi > 6. Vt. 8. /3y- 11. Vz: 

3. Vt 6. /3- 9 12 a 
° 5 ° 8 . 4/5 e /7 


13. The area of a garden plot in the form of a square is 1° 
square rods. Find the length of a side to two decimal places. 

14. A board in the form of a square contains 4,4 square feet. 
Find the length of a side correct to two decimal places. 

152. Meaning of simplification of a radical. The expres- 
sion under a radical sign is often called a radicand. Expressions 
involving square roots are, in general, said to be in simplest, 
form when they are reduced so that the radicand (1) is in in- 
tegral form, (2) contains no factor which is a perfect square, 
and (3) contains no radical in the denominator. 

For example, | 

VJ/27 = /9°3 = 3, 


Vite = Velney = ey Vay 


and 


VE = VE3 = V3. 


The right-hand forms of expression are simplest for purposes 
of computation. This fact will be appreciated if we compute 
the values of the different expressions to a certain number of — 
decimal places. 


? Arts. 152, 153] EXERCISES WITH RADICALS 229 


EXERCISES 


Given 1/2 = 1.4144, V3 = 1.732+, compute the following, 
correct to two places of decimals: 





Pw Aris: Bar a2: 5. »/108 + V5. 

an 1 se a 
2. /27. 4, et 6. /75 + V3. 
Simplify the following : 
7 +/50. 11. ./1 — (4)? 15. +/200a°0?. 

ne ce 

8. 1/20a2b. 12. ./2 + (#)?. 16. / ie (F) 
9. 3/5. 13. 4/1227; Le vise 
10. 5/2. 14. /63zy%2?. 18. 3,/2. 


Put each of the following in a form without a number 
written outside the radical sign: 


ALO ee 
Sotution: In this case, 2.\/3 = 4: /3 = V12. 





F oe ae 4 DS ak 
20. 5/3. 24. 32~/2" — y’. 27. y V3 Zt 
x ee 3 
21. 24/3. 20% >= * 28. 3/4. 
EV ea ivV% 
x /y ee ee 
- -/= 26. =~ 2. 29. —v/7°. 
22 =e 6. SV 9. Vi 
23. 3q./2p- 


153. Addition and subtraction of radicals. When two 
radical expressions, such as 3./a and 5,/a, have the same 
expression for a radicand or can be given the same radicand 
by simplification, they are said to be similar radicals. Similar 


230 RADICALS [Cuap. XXII. 


radicals can be combined into one term by additions and 
subtractions. 
Thus, V/8+ /32 = 2,/2 + 4,72 
=(2 + 4)v2 
= 6/2. 
Again, /4a + /9a + /l6a = 2V/a + 3a +4Va 
= 9V/a. 
EXERCISES 
Combine the following and simplify as far as possible with- 
out approximating roots : 


V18 + V50 + V2. 

n/ Loe wy ieee Toe 

/20 + »/45 — +/80. 
V/128 — V18 + V72. 

/108 — 147 + V/75. 
/700 — »/28 + +/63. 

/32a — /8a? + +/18a?. 
/32a — V/8a + V/18a. 

Va + 4a — /166 + 90. 

/d.— 0 +'./4a = 4b x/0a Ob 

. V(a — b)?m + V/9(a — b)*m — V/4(a — b)?m. 
. V3202b? + 1/12802b? + »/162026°. 

. V27 + V108 + V/45 + 80. 

V5 + -V125 + V9 4+ V 18. 

. W522 — 50x + 125 + ~/5a? — 10x + 5. 
OAV nk a VA PA EN Oy 

. 2/8 + 40/60 + V/15. 

EN AN BEN OG 


See Coa este sen ae ea Oa Ee 


en a Ss Se — = = 
ONDA FF WOW DY FH SO 


© Anns. 153, 154] QUADRATIC SURDS 231 


19. ~/4ab? + bv/25a + Va(a — 3b)?. 
20. 5x/300a3b?2 — 3+/243ab?. 


21. Find the sum of the lengths of diagonals of three squares 
of sides 10 feet, 20 feet and 30 feet. 


154. Multiplication of quadratic surds. To multiply one 
quadratic surd ~/a by another +~/b, we use the principle, 


Art. 138, : hs. 
Re bit iab: 
That is to say: The product of the square roots of two num- 
bers 1s the square root of their product. 
Example 1. Find the product of »/3 and ~/27. 
SoLuTION: +/3- 1/27 = +/3-27 = V/81 = 9. 
Example 2. Multiply 2\/3 + 3/5 by V5 - V3. 





SoLuTION: B2Ao 3/5 
Roe Ne 
2/15 +15 
34/15 \— 6 
~V/15 + 9 
Example 8. Multiply \/a +a —6 by Va - Va —b. 
SoLuTION: /a+Va—b 
N/a N/a. 0 
a + +a? —ab 
— v/a? — ab — (a — bd) 
a-a+b 
Hence, the product is b. 
EXERCISES 
Perform indicated operations and simplify results : 
fas V7. Lig PUAN EA 
2. /8-/12. 5. Va. V 2%. 


3. v/a - Vda. BEV se’. 


232 RADICALS [Cuar. XXL. 


TV 5-V 4. 9. Vrsit3 - ~/ rst. 
8. 1/2. 738. 10.--V xyz - / EI a 


11, 5 (34/2)= AV BBA Az 81 

12. (5V3 + V7)(5V3 — V7). 

13. (20/5 + 8/3) (4/5 = 5+/3). 

14. (Va + V/b)(Va — Vb). 

15. (3V/a + 2V/b)(5V/a — 6y/b). 

16. (VW2+ V3 + V5)(VW2 - V3). 

17. (R- SV2)(R + *y/2). 

18. Find the value of 2? if = 1/5 + V/10. 
19. Find the value of x? + 5z — 1 when x = 


—5 +4 Vv 29° 
2 

Slee 
3 

21. Find the value of 2? + 6a — 4 for x = a mS: 


22. Find the value of 32? + 32 — 5 when x = ; + us 


23. Does 1/3 + ~/2 satisfy the equation 2? — 4% + 1 = 0? 

24. Does —3 + V14 satisfy the equation x? — 6x — 5 = 0? 

155. Division of quadratic surds. In dividing one radical 
expression by another, the division is usually indicated by 
using the dividend as the numerator and the divisor as the 
denominator of a fraction. 


20. Find the value of 32? + 27 — 2 when z = 


ye 
Thus, to divide ~/a by ~/, we write VE 
We may also make use of the following principle: The 
quotient of the square roots of two numbers is the square root of the 
quotrent of the numbers. _ | 
That is, Ba Wie 
V/b 


Arts. 155, 156] RATIONALIZATION 233 


EXERCISES 


Express each of the following as a fraction under one radical 
sign and reduce to simplest form: 


v2. AGM A 
Neon * V8 ane 2 
ey. v5 V10 
"V5 " 4/20 " 4/30 


156. Rationalization of denominators. While we may 


Li 


' thus indicate (Art. 155) the quotient of two quadratic surds 


in two ways, it is frequently necessary to go further and present 


the result in a simpler form or find an approximate value of 
the quotient. 


Thus, it may be necessary to find the values yp se fea all to 


mae V5 - V3 


three places of decimals. 


To find the approximate value of Ye three methods are shown in 
Art. 151. It is a saving of work to multiply the numerator and denom- 
inator by the factor 4/3 so as to make the denominator rational. 
Pay bs 
V5 - V3 
square roots of 5 and 3, then subtracting the square root of 3 from that 
of 5 and performing the division, involves three rather long operations. 
The labor of two of these can be avoided by multiplying the numerator 


and denominator by V/5 + V/ 3. Thus, 


V5 V5 4 V8 5+ VIB _ 5+ V15 


To find the approximate value of by first extracting the 





eens Vorye 6-38 2 
Finding the value of sa ee wee involves but one long operation. 


The process just explained is called rationalizing the denomi- 
nator. 


234 


157%. Rationalizing factor. 


RADICALS 


[Cuap. XXII. 


The factor by which the terms 


of a fraction are multiplied to make a denominator rational is 
called a rationalizing factor. 
If the denominator is of the form +/a, then clearly ~V/a is 
a rationalizing factor. 
If the denominator is ~/a + VW b, then Va — Vb is a ra- 
tionalizing factor. 


EXERCISES 


Find equivalent expressions with rational denominators: 


1. 





Vig BVB42v8 
V5 Rew cyen«: 
Wee biel 
4/20 AY Die 
Va 5 es cae 
Vb "4/5 — V3 
Wala Richey 
a/3b2 4/3 — V2 
V8+V2 4) 8+V5. 
WN tea vat 





2/15 - 6 
V5 2N/2 
eee 
V5 EV/2 
5 +2V2 

4 — 24/2 
re 
Vine 
See . 
BV/5 — 2/3 


ik 


12. 


13. 


14. 


15. 


Find the value of the following to four places of decimals, 
avoiding as much lengthy computation as possible : 


9 V8+V5. 


1 





i 
Af Da 


V7 


a Re. 


18. 


8V3 + 2V2 
V3- V2 


6a 


20. 


5 +272 
42070 


1 ——] when « = V2. 


f 


> Arr. 158] EQUATIONS INVOLVING RADICALS 235 


22. If a regular decagon is inscribed in a circle of radius 7, 
one side of the decagon is (V5 —1). Find the ratio of the ra- 


dius to one side (correct to three decimal places). 

158. Solution of equations involving radicals. Equations 
involving radicals may sometimes be conveniently solved by 
squaring each member. This operation is justified on the 
ground that each member is multiplied by the same number. 


Example 1. Solve Vx —1 =5. 
SoLuTion: Square both members and we have, 
x —1 = 25. 
Hence, x = 26. 
CHECK: V/26 -—1 =5. 
Example 2. Solve Vz -1+ /x—4 =2. (1) 
SoLution: Transpose one of the radicals, say ~a — 1, then 
We aaa 2 — Vx —, (2) 
Square both members, eet A Aas = Pee = 1s iG) 
Simplify, Av/az — l= 7. (4) 
Square both members, 16(2 — 1) = 49. a 42/45) 
Hence, 16x = 65, 
a = 4y5. (6) 
CHECK: /4}, —-4 =2- A/a Ls 
or + —=2 -—1} 


ee 
paale ' 


Note that if radicals are involved, it is well to get a radical 
alone on one side of the equation before squaring. It is spe- 
cially important that all results be checked by substitution 
when the members are squared in the course of the solution. 
This is necessary for the reason that solutions may be intro- 


236 RADICALS : [Cuap. XXII. 


duced by the operation of squaring both members which do not 
satisfy the original equation. 

Thus, given x = 5, (1) 
let us square both members. | 
This gives 8 25, (2) 
Extracting the square root of both members, 

aim 

But x = —5 does not satisfy equation (1). 


EXERCISES 
Solve and check results: 
1\/ 2210 


2. Wx+10 = VW 22 — 6. 
38. 24-5 = V2 — 4¢ + 23. 
4 


— Ve— 74 4+4=V2+2— 90, 


6 Ve t+1+ V2 —6 =7. 
6 VWe+9=14+V2—5. 
7 VePX+_7+2=9. 

8 6 — Va = Vx +10. 

9 Vy 420 —/7 a fo 


10. V2+543 = 8-4/9 


ct Vid A yen 
Bobs Vata=Va2 +a. 

13. \/ 2e = 8a 44/22 = 8a7u 
14. V/4a+¢ =2Vb+2e4+V-2. 


© Arr. 158] EXERCISES AND PROBLEMS 237 
Vat4 x+4 v/x + 20 z+ 20 
V2e+1 V2e+10 


CE Wee 
J ie Sara. 


17. V/0+-2-— 716 42 = 0. 
18. VWr4+25=14+ v2. 


19. The time ¢ in seconds for the complete vibration of a 
simple pendulum is given by 


15. 





ae} 


ea 
™ V 39.9" 


where | is the length (in feet) of the pendulum, and 7 = 3.1416. 
Solve the equation for / in terms of ¢ and find the length of a 
_ pendulum that makes a complete vibration in two seconds. 

20. The velocity of a falling body starting from rest is 
given by v = ~/ 2gs, where s is the distance passed over and 
g = 32.2. Express sin terms of v and g, and find the distance s 
that corresponds to a velocity of 128.8 feet per second. 


CHAPTER XXIII 
QUADRATIC EQUATIONS 


159. Quadratic equations solved by factoring. In Art. 84, 
we have solved by factoring some special equations of the form 


ax? + bu +c = 0, 
and have learned to call such equations quadratic equations. 
Take, for example, the equation 
2a? + 5a —3 =0. 


The factors of the left hand member are easily found. They are 2x -1 
and « + 3, and we may write our equation in the form 


\ (2x —1)(x@ +3) =0. 
Any value of x that makes either factor zero satisfies the equation. 
If x = 3, we have , 
(2-2 -1)@ +3) =0-( +3) =0. 
Again, if « = —3 we have 
[2(-3) -1](-3 +8) = [2(-8)-1]-0 =0. 


Hence, ¢ and ~3 are solutions or roots of the given quadratic equation. 


EXERCISES 

Solve the following equations: 

1. 2? — 7a +6 =0. 7. 22 +4 = 3 = 
2 2 BS 5 =U, 8. 3a? — 2x — 8 = 0. 
3. 6a? — 1387 + 6 = 0. 9. 67? + 352 — 6 = 0. 
4, 3x? — 5x = 0. 10. 377+ 74 +2 =0. 
5. 2y? — y — 36 = 0. 11. 8? + 2-1 =0. 
6. 7s? + 10s — 8 = 0. 12. 227 — 7x — 15 = 0, 


238 


“Ants. 159, 160, 161] COMPLETING THE SQUARE 239 


13. 8? —10¢-—3 = 0. 17. 2y? —7y +3 =0. 

14. 2? — 52 = 52 — 25. 18. 9 — 5s? — 12s = 0. 

15. 2? — 227 — 48 = 0. 195 2572 — G0 307 
16. 2? — 207 + 51 = 0. 20. 277 — x = 3. 


160. Completing the square. 


From 
b\* b\? 
(s+5)-e+b2+(5), 


2 
we note that the last term (5) is the square of half the co- 


2 


efficient of x. Hence, the expression 


x? + bx 


2 
lacks only the term (5) of being the square of « + z 


Therefore, if the square of half the coefficient of x be added to 


an expression of the form x? + bx, the result is the square of a 


binomial. 
Such an addition is usually spoken of as completing the 
square. 
EXERCISES 
Complete the square in each of the following : 
1. 2? + 42. 8. 2? 4 122. 5. x? + 9a. 
oe 2 82. 4, 2? + 32. 6. 24+. 
7. «x? — 2az. 
Hint: Make b = — 2a in the above discussion. 
8. x? — 4a. 11. (27)? + 4(2x). 14. 162? + 8z. 
9. x? — 32. 12. 4a? + 82. 15. 92? — 122. 


10. (2x)? + 2(2x). 138. 162? — 2(47). 16. 92? + 92. 


161. Equations solved by completing the square. Equa- 
tions of the form az? + be +c¢ =0 


240 QUADRATIC EQUATIONS  [Cuap. XXIII. | 


may be solved by a process that involves completing the square 


on ax? + ba. 
Example 1. Solve the equation x? — 6x — 7 = 0. (1) 
SoLuTion: ‘Transpose —7, and we have 
v— tr =7. (2) 
Add 9 to each member to complete square in left member. 
This gives xv -—-6r4 +9 =16. (3) 
Taking the square root of each member, 
zr-3=+4. 
Hence, zx =7or —I: 
Check by substitution in (1). . 
Example 2. Solve the equation 
2y? + 84 +1 = -2e +4. (1) 
SoLtuTion: Transpose and divide each member by 2, 
a + x =} (2) 
Add 73 to each member to complete square in left member. 
This gives m+ 30 +43 = $3: (3) 
Taking square roots, zr+%=+F: (4) 
Hence, x = ¥, (5) 
and also r= -—3 (6) 
Check by substitution in (1). 
Solve and check : Wage ea 
1. 2? — 37 — 2 = 0; 7. x? + 27x + 140 = 0. 
2. 2 —67%+4=0. 8. n(n — 1) = 210. 
S$... 3? — §—4= 0). 9. y® — 10y = 75. 
4. x? — 6x = 40. 10. —8 = 2x? + 102. 
5. 2? — (x 6 = 0; 11. ¢t(¢+ 4) =7. 
6. 32? — 1074+ 3 = 0. 12. (n+ 1)? — 8(n +1) = 16. 


162. Solution by Hindu method of completing the square. 
In case the coefficient of a? in the equation is not unity, as in 


2a? — 42a — 7 = 0, 


both members may be divided by this coefficient to obtain 
an equation in which the coefficient of 2? is unity, and the 


equation may be solved as shown in Art. 161. 








| 
7 


» Art. 162] HINDU METHOD 241 


However, the following method sometimes avoids the 


7 introduction of fractions until the last step of the work. 


Example. Solve the equation 


22? — 4x = 7. (1) 
Multipy each member by 8 (four times the coefficient of 2?). 
This gives 16x? — 32a = 56. (2) 
or (4x)? — 8(4x) = 56. 
_ Complete the square, 16x? — 32% + 16 = 56 + 16 = 72. (3) 
4 Extracting square roots, 4x —-4 = +/72 = 4+6vV2. 
* Hence, z= 1+ anes (4) 


Check by substitution in (1). 


} 


It should be noted that the number 16 added to complete 
the square is the square of the coefficient of x in the original 
equation. This method of completing a square on aa? + bx 


_ after multiplying by 4a is known as the Hindu method. 








EXERCISES 
Solve and verify by substitution in each case : 
1. 52? — 3a —2 = 0. 16. 3s? + 4s = 95. 
2. s* + 2s = 120. 17.. Q2 — 3) = 62 +-1L. 
8. x2? + 227 = —120. 18. m(m +4) = 7. 
4, 2? —111+ 28 = 0. Lover Jer = 1: 
5. 2n(n + 4) = 42. 20. 15a? — 1474 + 3 = 0. 
6. 62? — 52 —6 = 0. 21. 2? +67+5=0. 
ot — OL = 8. - 22. 22° — (= 42. 
8. 2m? + 3m = 27. 23. s? +12 = 8s. 
8.0.22" + 0.92 = 3.5. 24. 37? +r = 200. 
| 10. 03x -O7e=1. 26. + if =2. 
11. 42? — 192 = 5. 26. (x +1)? — 8(x +1) = 16. 
: 12. 2s? — 5s = 42. 27. 3(v? — v) = 2v? + 5v 4+ 4. 
(13. Se — 14t = -8. 28, s = 4542. 
tat + 27 = 32. 29. 8m — 10 = m?. 
15. 18v? + 6v = 4. 80. $x — $07 +2 =0. 


242 QUADRATIC EQUATIONS [Caap. XXIII.” 


163. Type form of a quadratic equation. The typical form 
of a quadratic equation is 


ax? ++bx+c=0, (1) 


where a, 6, and c do not involve z, and may have any valuce 
with the one exception that a may not equal zero. 

Since the result of multiplying the members of an equation 
in this typical form by any given number is an equation in the 
typical form, the a, b, and ¢ can be selected in an indefinitely 
large number of ways. 


EXERCISES 


Arrange the following equations in the typical form 
ax? + bx +c = 0, and indicate the values of a, 6b, and c in the 
resulting equations: 3 


5 4a? — 5 + 50 = Sn? +m. 


3 
Soutution: By transposing and collecting terms, 
ae +32 —(5+m) =0, 
from which a = 4°, b = 3, c = —(5 +m). 
2.92 (me bt 8. (2x — 3)? = 67 4+ 4. 
= 2 2 
4. 22+ (ax + 6)? = 0. 10. 1547+ 2 =1 0 
5. 3a? + 52 — 7 = x? — 2a. 11. 1 -—2Qy+y* =2. 
6. (y — 3)? + 4y = 16. 12. 30? — 7x = a(x + 1). 
7. aa? + 2eu = —d. 13. (27 — 3)? — 64 +1) =38: 


14. r(v7 +4) = 7. 
15. 4m2x? + 3k? — 8mx + 82 —-m+k =0. 


164. Quadratic solved by formula. The process of ‘com- 
pleting the square’’ when applied to the typical quadratic 


az? +ba+c=0 





i 
i] 
} 
f 
; 












| 


Arr. 164] SOLUTION OF QUADRATIC 243 


leads to a very useful formula which may be used to solve all 


quadratics, and which affords the most convenient method of 


_ solving many quadratics. 


To obtain the formula that gives the solution of 
ax? + br +c = 0, (1) 
transpose c and divide by a. This gives 


A b Cc 
e+ -7 = —-. 
a a 
4 
Add (5) to both members to make the left-hand member a 


perfect square. Then 








b b2 Cc b? b? — 4ac 
2 = ae wes 
ae ae a7 4q? a 4a? 4q2 ? 
( 6b } _b- 4ac 
a a ATi antag ye 
. ie 
Extract the square root, x + a = + Vb = dao 
. 2a 2a 
—b+ 4/ b? — 4ac 
or 4 (eee ee 
2a 


Therefore the roots of the general quadratic equation, 


: az? + br +c = 0, 


are 
— b+ a/b? — 4ac 
2a 
and 
—b — v/ b? — 4ac 
2a 


as may be verified by substitution in equation (1). 
—Bb+ 4/ b? — 4ac 


he expression ~ 
Bae EXD 2a 


| may therefore be used as a formula for the solution of any 


quadratic equation. 


[Cuap. XXUI._ 


244 QUADRATIC EQUATIONS 


Thus, to solve 
27? —4¢ —7 =0, 





we substitute in the formula, a = 2, b = —4, ¢ = —7, and obtain 
path Vib = 432m 
4 


are the roots of the equation. 
These values of x are identical with the solutions given in (4), Art. 162. 


EXERCISES 


Solve the following equations by the use of the formula, 
and verify by substitution: 


1. 152? — 1474 +3 =0. 16. 227 = 9 — 3z. 
2. 6x2 = 192 — 10. 17. x(2¢ +3) +1=0. 
3. 72 — 24/382 +2=0. 18. a2 = —4(4 —3). 
4. 1627 — 347 + 15 = 0. 19. 4(2x + 5) = 2. 
5. (2x-+ 5)? = 452. 20. 4 = «(3x + 2). 
6. 2a +7 = = 21. 422 — 32 —2 = 0. 


7. 3° +7 = 200. 
8. 92? + 32 = 2. 


9. 62? + 5” = —1. 


. 2(22 +3) +1 =0. 
. 1 sae 











2 3 
2 = 
10. 722 + 2a = 32. 5. ee 
a4 4 
1 PO8E 27 — 2 ee 
LAE Tt thea eer 16. ie +5 
12. 2y? — 5y — 150 = 0. 27. 2s? + 3s = 27. 
13. 42? — 17x = —4. 28. (2x — 3)? —- 6(4 +1) +15 = 0. 
14. 47 -27 -3 =0. 99. 22+ (5 — 2)? =e 


15. 322 + 7x = 110. 


~t+V24+6 = 14. 


Hint: See Art. 158. 


‘é Arr. 165] QUADRATIC EQUATIONS 245 


165. The special quadratic aw? +¢= 90. When b =0 in 
the typical quadratic equation, the solution is very simple. 


Thus, from az* +c = 0, (1) 
we have Ci t= 6, 
Y= ay 
a 


and X= Ves 
a 


Equations of the type of (1) are solved in Art. 139. 


MISCELLANEOUS EXERCISES AND PROBLEMS 


Solve the following equations : 




















ia — Liz + 30 = 0. 16. 4s? + 12s+5=0. 
2. (x? + 22 = 32. 17: a2? 4-22 4.='(), 
3. 2? — 127 = 28. 18. 427 — 287 + 49 = 0. 
4, x? — lor = 0. 19. 6— 3x2 — x? = 0. 
5. 27 — 25 = 0. 20. 64 + 2? — 162 = 0. 
. + Deen et A. 
6. a? — 4.37 + 3.52 = 0. oh Vp ly ea Lea 
oe — 0:25 =.0.15. B27 GF — 350 - G =.0. 
= 8. s* — 123s.+.27 = 0. 23. 72+ 10¢ —8 = 0. 
2 
pee? Log 24. 20s? + 11s — 3 = 0. 
4 3 
* 4 ee+2 «24-5 
ae 5 tO 2b: Dees ea Se 
41, 42? — 282 + 45 = 0. 26. ay eg) 
t-—3 x 
1l+2 2-1 
2 “Ss ee = ie pesieaa my eer ie AS 
12, 7 +2-@-a=0. 61 -FSap Sar ee 
13. 6y? + 35y — 6 = 0. 28. 0.42? — 0.27 — 0.2 = 0. 
14. 1522+ 162+ 4 = 0. 20 sept Dba 1 5G al), 


15. 67? + 5¢ = —-1. 30. (1 — e”)a? — 2mz + m = 0. 


246 QUADRATIC EQUATIONS  [Cuar. XXIII. & 


31. The sum of two numbers is 30, and their product is. 
176. Find the numbers. 


SOLUTION: 
Let x = one number. 
Then 30 — x = the other. 
Since their product is 176, 
and 2(30 — x) = 176. 
Solving the quadratic, x = 8, or 22, 
and 30 — x = 22, or 8. 


Hence, the numbers are 8 and 22. 


32. Divide 50 into two parts whose product is 600. 

33. Divide 8.4 into two parts whose product is 17. 

34. Find two consecutive integers whose product is 72. 

35. Find two consecutive integers the sum of whose squares 
is 145. 

36. A square field contains 10 acres. What are its dimen- | 
sions? | 

37. A rectangular field is two rods longer than it is wide 
and it contains 6 acres. What are its dimensions? 

38. Some boys are given 3500 square feet of land in rec- 
tangular form for basket-ball grounds. If the grounds are 20 
feet longer than they are wide, what are the dimensions? 

39. The dimensions of a picture inside the frame are 14 by : 
18 inches. What is the width of the frame if the area of picture 
with frame included is 320 square inches? 

40. A piece of tin in the form of a 
square is taken to make an open box. 
The box is made by cutting out a 3-inch 
square from each corner of the piece of 
tin and folding up the sides. (Fig. 34). 
The box thus made contains 192 cubic 
inches. Find the length of the side of the 
original piece of tin. 

41. A farmer has a 10-acre wheat field in the form of a 
square. In cutting the wheat, he cuts a strip of uniform 


Fig. 34 


|. Anrs. 165, 166] | QUADRATIC FUNCTIONS 247 


" 


~ width around the field. Find the width of the strip when one- 
half of the wheat is cut. 


42. A stream flows at the rate of 5 miles an hour; a crew 
rows 8 miles down the stream and back to the starting point in 
4 hours and 40 minutes. What is the rate of the crew in still 
water? 

43. The distances through which a body falls in different 
periods of time are to each other as the squares of those times. 
In how many seconds will a body fall 400 feet, the space it 
falls through in the first second being 16.1 feet? 

44. A and B distribute $1200 each among some poor people; 
A gives to 40 persons more than B, but B gives $5 more to 
each person than A gives; find the number of persons helped by 
A and by B. | 

45. A rectangle is 15 by 20 inches. How much must be 
added to the length to increase the diagonal 3 inches? 

46. One leg of a right triangle is 6 feet, and the other leg 
is one-half the sum of the hypotenuse and the given side. Find 
the sides of the triangle. 

47. A rectangular park 56 rods long and 16 rods wide is 
surrounded by a street of uniform width. This street contains 


4 acres. What is the width of the street? — 


48. The circumference of the fore wheel of a buggy is 3 
feet more than that of the hind wheel. If the fore wheel makes 
125 more revolutions than the hind wheel in going a mile, find 
the circumference of each wheel. 

49. What is the area of a square whose diagonal is one foot 
longer than a side? 

50. An automobile round trip of 250 miles was made in 11 
hours. On the return portion of the trip, the speed was 4 miles 
an hour more than on the outgoing portion. Find the rate each 
way. 


166. Graphs of quadratic functions. Any quadratic func- 
tiion, say 3a?-—5x%-—2, may be represented graphically 


| 


248 QUADRATIC EQUATIONS [Cuap. XXIII. 


(Fig. 35). The graph may throw considerable light on the solu- 
tion of the equation | 
3x? — Sa — 2 = 0 
formed by equating the quadratic function to zero. 
In order to plot the graph of 32? — 5a — 2, we first fen a 


table of corresponding values as follows (compare Arts. 18, 120): | 


To represent the numbers in the table conveniently, it is 
best to use different scales for x and for the function 
3a2 — 5a — 2, since for a range of values from —5 to +6 on 2, 
the function takes values from —4 to +98. Plotting points from 
our table of values and drawing a smooth curve through these 
points, we have the graph in Fig. 35. 


- oa — ba —2 























| 


oe CoP WNW KH © 
ar 
= 




















ae ech lee Leet 
arP WN O 
On 

bo 

S 














y’ Horizontal - 2 spaces = 1 unit f ; 


Vertical - lspace = 4 units 





S 





Arts 166, 167] IMAGINARY NUMBERS 249 


It should be observed that the graph crosses the X-axis at 
two points. The values of x that correspond to these points 
make the function zero, and are therefore the roots of the 
equation formed by equating the function to zero. The 
function becomes zero both when x = — 3 and when @ = 2. 


EXERCISES 


1. Solve the equation 32? — 5a — 2 =0, and explain the 
meaning of the roots by the use of the graph (Fig. 35). 

2. To ask for the values of x that make 32? — 5x — 2 equal 
to 10 means what on the graph? 

3. Form the equation whose solution answers the question 
in Exercise 2. 

Solve this equation. 

Plot the graphs of the following functions : 


4. a? — 32 + 2. 7. 244743. 
5. aw? — 7x + 12. 8. 32? + 7x + 4. 
6. 277 + 32 +1. 9. 2? — 5a + 4. 


167. Imaginary numbers. Certain quadratic equations, for 
example, 
z?+1=0, and a —-6%+15=0 
demand for their solution an extension * of our number system 
to include the square roots of negative numbers. From the 
equation x? + 1 =0, we have 


7 = |, 
and the equation may be said to ask for a number whose square 
is —1. It is useful to create such a number and it is customary 


to write it+/—1 or i. That is, 7 is to be thought of as a 
number whose square is — 1. The square roots of negative 


* For other extensions, see Arts. 1, 2, 21, 148. 


250 QUADRATIC EQUATIONS 


[Cuar. XXIII. 


numbers are called imaginary numbers, but we shall see in the 
chapter devoted to these numbers in the advanced course that 


this designation is a misnomer. 


It is usually convenient to re-— 


duce any imaginary number to the form az, where a is a real 


number. 


EXERCISES 


Reduce the following numbers to the form az: 


1. VY -4. 


Soutution: VW —~4=Y%—1-4= VW4-VY—1=2:-V -1= 2%. 


TH ee 


Bye 


2. V/—-9. 
$247 216) 


7 ey Ga 9. ~/—a?. 13. VW—(2 + @) 
rey ea 
6..4/—5 10. ~/ —2c. 


Solve the following equations : 


14. 22+ 1 = 0. 17.. 22° 4 eee 
15. 2?°4+4= 0. 18. 3274+ 2 = 0. 
16. 277 + 6 = 0. 


168. Graphical meaning of imaginary roots. 


a). 


It will be 


instructive at this point to note an important property of the 
graphs of quadratic functions that give equations with imagi- 


nary roots when the functions are equated to zero. 


| Arr. 168] IMAGINARY ROOTS 251 


To illustrate, plot the graphs of 


x? — 67 +C 


(1) when C = 0, 
(2) when C = 5, 
(3) when C = 9, 
(4) when C = 15. 





Wes Horizontal - 2 spaces =1 unit 
Vertical - Ilspace =1 unit 


Fia. 36 


The four functions plotted differ only in the value of C. They are 
similarly shaped graphs; but, as C is increased from 0 to 15, the graph is 
simply moved upward on the paper. From the graph for C =0, the roots 
of the equation 2? — 6x = 0 are seen to be 0 and 6. From the graph for 
C = 5, the roots of x? — 6x +5 = 0 are seen to be 1 and 5. 

We note that the graph for C = 9 merely touches the X-axis at the 
point x = 3. | ) 

The graph for C = 15 does not at all intersect or touch the X-axis. 
_ This is the case with the graph of the function when the roots of the equa- 
_ tion formed by equating the function to zero are imaginary numbers. 


252 QUADRATIC EQUATIONS [Cuar XXIII. 


To illustrate, solve the equation 


a —6¢ +15 =0. (1) 
Weatad oe 
=3+ V-6, or3 +i V6. (2) 


These solutions are imaginary. 

Example. Operating with 7 as with any other number, and remember- 
ing that 72 = — 1, verify by substitution in (1), that 3 +7 V6 and 3-1 V6 
are solutions of the equation. 

Thus, 

(3 + 1/6)? — 6(3 +iv6) +15 =9 4 6/6 + 6? — 18 - 6ivV/6 +15 
=9-6-18 415 
= 0. 


EXERCISES 


Plot graphs of the following functions and solve the equa- 
tions formed by equating these functions to zero: 


Lotte 6. x — 4a. 

2. x? — 34 + 2. 7. 32? + 5x — 2. 

3. vw — 4443. 8. 3x7 + 5x. 

4. x? — 474 4. 9. 3a? + 5a + 4. 

5. 2? — 4x 4+ 10. 10. 42? + 32 — 1. 
PROBLEMS 


1. The difference of two numbers is 5, and the sum of 
their squares is 325 ; what are the numbers? 

2. The altitude of a triangle is 6 inches more than its 
base, and the area is 108 square inches. What is the 
altitude? 

8. Divide a line 386 inches long into two parts such that 
the rectangle whose sides are equal to the two parts has an 
area of 315 square inches. 


Arr. 168] PROBLEMS 253 


4. The perimeter of a rectangle is 24 inches, and its area 
35 square inches. Find the length and breadth of the rectangle. 

5. The perimeter of a rectangle is 14 inches and the area 
is 25 square inches. Find the length and breadth of the rec- 
tangle. Explain why this solution is imaginary while that in 
Problem 4 is real. 

6. A square pond is surrounded by a gravel walk with a 

uniform width of 2 yards. The area of the walk is equal to that 
of the pond. Find the dimensions of the pond. 
7. To get from one corner of a college quadrangle (rec- 
_tangular) to the opposite corner, I must go 140 yards, around 
the sides ; if I were allowed to cut diagonally across the grass 
I should save 40 yards. What are the dimensions of the 
- quadrangle? 

g. At what price per dozen are eggs selling when, if the 
price were raised 5 cents per dozen, one would receive twelve 
_ fewer eggs for a dollar? 

9. The sum of the base and altitude of a triangle is 6 inches, 
and its area is 10 square inches. Find the base and altitude. 
Explain why the results are imaginary. 

10. A certain man has an income of $5000 more than his 
exemption of $3000 from income tax. After deducting a per- 
centage for federal income tax on the $5000, and then an equal 
percentage from the remainder for a state income tax, the 
income is reduced to $7900.50. Find the rate per cent of the 
income tax. 

141. A man bought a number of $100 shares of R.R. stock, 
when they were at a certain rate per cent premium, for $3500; 
and later, when they were at the same rate per cent discount, 
sold them all but 5 for $1200. How many shares did he buy, 
and how much did he pay per share? 

12. The telegraph poles for a certain line are set at equal 
distances. If there were 4 more per mile, the distance between 
them would be decreased by 66 feet. Find the number of poles 
per mile. : 


254 QUADRATIC EQUATIONS  ([Cuar. XXUIL. . 


13. A beginner wishing to simplify (x + 5)(a — 2) just leaves 
out the parentheses and writes x +5a2—2. Are there any values 
of x for which the two expressions are equal? 

14. If a ball is thrown upward with an initial speed of v 
feet, per second, it is known that after t seconds its height will 
be ot — 16.1¢ feet. If such a ball is given an initial speed of 
100 feet per second, after how many seconds will it be at a 
height of 100 feet? After how many seconds will it return to 
- the starting point? 

15. A balloon is 1 mile from the ground and is descending 
at the rate of 5 feet per second when a sand bag is dropped. 
If the formula 5¢ + 16.1@ gives the distance that the bag will 
fall in ¢ seconds, find the number of seconds required for the 
bag to reach the ground. 

16. A farmer is plowing around a field 160 rods long and 80 
rods wide. How wide a strip must he plow around it to 
make 10 acres? Draw a.diagram of the field and verify 
your result. 

17. Find the side of a square field whose area is equal to that 
of a rectangular field whose length exceeds a side of the square 
by 40 rods and whose width is 20 rods less than a side of the 
square. : 
18. The base of a triangle is 6 longer than the altitude, and 
the area is 176. Find the base and altitude. 

19. In a right triangle, the hypotenuse is 80 inches, and 
one leg is 16 inches longer than the other. Find the dimensions. 

20. Divide the number 20 into two parts whose squares 
are in the ratio 4 to 9. 

21. An open box is made from a square piece of card-— 
board by cutting square pieces out of the corners and then 
folding up the flaps. Find the size of cardboard that is used 
to make a box 4 inches high to contain 144 cubic inches. 

22. From a cardboard a box twice as long as wide and 
containing 240 cubic inches is made by cutting 5-inch squares 
from the corners. Find the dimensions of the cardboard. 


Arts. 168, 169] PROBLEMS 255 


23. A stream flows at the rate of 5 miles per hour. A crew 
‘can row 6 miles with the stream and the same distance back 


‘in & hours. What is the rate of the boat in still water? (See 


Problem 32, p. 204.) 

24. A crew can row upstream against a current of 2 miles 
an hour, for a distance of 10 miles upstream and back again 
in 22 hours. What rate should we expect the crew to make in 
still water? 

25. The numerator and denominator of a given fraction 


‘together equal 80. If we increase the numerator by 8 and 


decrease the denominator by 8, the resulting fraction is $ as 
large as the given fraction. What is the fraction? 


f 169. Historical note on quadratics. Problems that involve quad- 


‘ratic equations were solved by Diophantus, the Greek algebraist, who 
lived in Alexandria about 300 A.D. But he gave only one value of the 
unknown. He did not recognize the meaning of a negative result, although 
he solved some quadratics by a method not unlike that of completing the 
‘square. 

When Diophantus came upon an equation both of whose roots are 
negative, he rejected the equation as absurd or impossible. When only 
one negative root occurred, he merely rejected that root. When both 
roots were positive, he took only the root that would be obtained by taking 
b+ V0 — 4ac 
+4. In 

2a 

fact, Diophantus looked upon a quadratic equation as having either one 
or no root. 

About five or six centuries after Diophantus, the Hintlus solved quad- 
ratic equations, and observed that they have two roots. They did not 
regard an equation as absurd because its roots are not positive, but merely 
rejected the negative rcots on vague grounds illustrated by the following : 
In solving the equation 


the positive sign before the radical in the formula 


a — 452 = 250, 


Bhaskar gives x = 50 and x = —5 for roots, but he says, ‘‘ The second 


value is in this case not to be taken, for it is inadequate; people do not 


approve of negative roots.” 

The Hindus did, however, observe that negative numbers may be 
taken to relate to debts if positive numbers relate to assets. It was not 
until the work of Descartes (see p. 181) became known that the theory 
of the quadratic was well understood. 


CHAPTER XXIV 
SYSTEMS OF EQUATIONS INVOLVING QUADRATICS 


170. Introduction. The typical form of a quadratic equa- 
tion in two unknowns is 


ax? + bry + cy? +dx+ey+f =), 


where at least one of the coefficients a, 6b, or cis not zero. Ex- 
amples of quadratic equations in two unknowns are 
x? — 2ry + dy? — 11 = 0, 
T+ ay +203 =0, 
sy —4=0. 


One such equation is satisfied by an indefinite number of pairs 
of values of « and y. For example, each of the pairs of values 
(0, 9), (1, 7), (2, 6), (4, 5) satisfies the equation, 

xy — 34 + 2y — 18 = 0, 


as can easily be shown by substitution. 


EXERCISES 

Show that each of the following equations in two variables 
is satisfied by the pairs of numbers given: 

1. czy — 34 + 2y —18 =0, (10, 4), (-14, 2), (—6, 0), 
(-3, —9). 

2. 2x oe 2xy a y? eo 1. 2 = 0, ae 1), (Is —4), (0, —2), 
(0, 1). 

3. x +y’ ot 0, (0, BS (0, —1), (3, 5), (-3, —$), (7s, V3). 

4. is x + 44 —3= 0, (0, 3), GL 0), (2, or 1), (3, 0), 
(2+ V3, 2), (2-V, 2). ) 


256 


_ Arts. 170, 171,172] SIMULTANEOUS QUADRATICS 257 


Find at least four pairs of values of x and y which satisfy 


~ each of the following equations : 


5. at+xex—y—6=0. 


SoLution: Substituting x = 1 in the equation, there results 1 + 1-y 
—6 =0,ory = —4. Hence, x =1, y = —4, satisfies the equation. Again 
substituting x = 2, we find y =0. Hence (2, 0) satisfies the equation. 
In this way any number of pairs of values satisfying the equation can be 
found. 


yy +22e%-—-y-2=0. 
2x? + dy’ = 4. 
etaeayty+2e4-—y—2=0. 
9. 2 +2742 =(44+1) (y - 2). 


PID 


171. Solution of simultaneous quadratics. Although there 
is an indefinite number of pairs of values of x and y which 
satisfy one quadratic equation in two unknowns, yet there are 
never more than four pairs which satisfy two different equations. 


_ For example, the four pairs of numbers, (3, 4), (—3, 4), ‘Gamer ay 


_ (—8, —4) satisfy both the equations, 


162? + 27y? — 576 = 0, 
e+y? — 25 = 0. 


No other pairs of numbers can be found which will satisfy both 
of these equations. The general problem in systems of simul- 
taneous equations involving one or two quadratics is the finding 
of all the pairs of numbers which satisfy both equations. This 
problem is in general quite difficult, but there are some types of 
these equations which can easily be solved. A more extended 
discussion, together with the graphical interpretation, will be 
given in the second course in algebra. 


172. One equation linear and one quadratic. A system of 
two equations in two unknowns in which one equation is linear 
and the other is quadratic, can be solved by the method of 
substitution. In this case there are in general two solutions. 


258 EQUATIONS INVOLVING QUADRATICS [Cuap. XXIV. > 


EXERCISES 


Solve the following pairs of equations : 
1227 — Ff = OY, 
ee ay te 


SoLtution: From the second equation, we find 
ei Te 2y, 
Substituting in the first and reducing, we have 
10y? — 69y + 98 = 0. 
Solving this equation for y we find 


y =2 or 4.9. 
Substituting these values in the second equation we find 
x =3, or —2.8. 


The solutions of the two equations are then 
(3, 2), (—2.8, 4.9). 
CHECK: (coe iia 
3 4+2.2 =7, 


2. (—2.8)? — (—2.8)(4.9) = 6- (4.9), 
{ BeBe bp re Wh) rc 


2. 27+ 47% — 25 = 0, 6. cy = 15 
ty ea: rt+y =2. 
Be ty la a ee 7. 237 “yy? ace 
se a 2x +y = 9. 
4.0 Py = 2; 8. XY == eae 
A rn VP mmps x+ty=dsl. 
5. Seale 9. LY —=.S08 
2x +y = 3. x — 2y =7. 
10.%2yi— 2 = 0, 
3x — y = O. 
11. zy + 7y + 6x — 38 = 0, 
x+y =. 


12. 20? — 3rzy — y’? = 1, 
64 + y = 3. 


| Arrs. 172, 173] SIMULTANEOUS QUADRATICS 259 


ie yr? =. 0(x + y) + 2, 
ee 
Dike 


14. 2(e + y)? — (w@ + y)(@ — 2y) = 70, 
2(a + y) — 3(@ — 2y) = 2. 


15. The sum of two numbers is }%. Their product is 3. 
What are the numbers? 

16. If the figures in a number of two digits are reversed, the 
new number is 27 greater than the given number. The product 
of the two numbers is 1300. What is the given number? 

17. The difference between the numerator and the denomi- 

- nator of an improper fraction is 4. If 2 be added to both 
- numerator and denominator the resulting fraction is less by 3's 
than the original fraction. What is the fraction? 

18. The diagonal of a rectangle is $ yards. The perimeter 
is 2 yards. What are the dimensions of the rectangle? 


173. Equations containing x? and y? only. The type form 


of this case is 
ax? +cy2+f =0. 


If, instead of x and y, we consider x? and 'y? as the unknowns, 
the method of solution is that for linear equations. In general 
there will be four solutions for a pair of equations of this type. 


Solve: EXERCISES 
or y= 2, 
xv? — 2y? = —4l. 


Soutution: Solving for x? and y? we find 2? = 9, y?= 25. 

Hence x = + 3, y = +5. The four pairs of numbers, (8, 5), (—3, 5), 
(3, —5), (—3, —5) will be found upon substitution to satisfy the two 
equations. 


2 
OE & = 20, 

fo ay = 11, : : 
x? 4+ Dy? = 22. 28 OE Ean 9 


260 EQUATIONS INVOLVING QUADRATICS [Cuap. XXIV. |. 


4°32" = by? = 3; 6. 22? — dy? = 6, 
xv? + y? = 25. 3x” — Qy? = 19. 

5. a+y* = 16, 7. ax? — by =a + BD, 
4x? + 25y? = 100. ba? + ay? = a + B? 


2 
8, ne 4% — 2( ~— +3), 
m m 


v f = 2 
mae 

9. The square of the diagonal of a rectangle is 34. _ If 
two such rectangles were put end to end, the square of the 
diagonal of the new rectangle thus formed is 109. What are 
the dimensions of the first rectangle? 


10. Two rectangles of the same size and having diagonals 
2% feet long, are placed end to end. The square on the diag- 
onal of the new rectangle thus formed is 5; square feet greater 
than the square on the diagonal of the rectangle formed by 
putting one of the rectangles above the other. What are the 
dimensions of the first rectangles? 


174. Special methods. The solution of two quadratic 
equations in two unknowns is usually in the nature of a puzzle 
for which no special rules are given. Often no solution can 
be found without the use of mathematics beyond the courses 
given in high schools. Many systems may be solved by 
special devices in which the aim is to find values for any two of 
the expressions « + y, x — y, and xy, from which the values of 
x and y may be obtained. Various manipulations are performed 
in attaining this object, according to the form of the given 
equations. 

Whatever method is used in solving simultaneous equations, 
it must be kept in mind that the ultimate test of a solution is 
substitution in the given equations. 


\ Arr. 174] EXERCISES 261 


Example 1. Solve the system 


ev +-xy = 12, (1) 
y + zy =4. (2) 
SOLUTION: 
Adding (1) and (2), v+2ry +y? = 16, (3) 
whence x+y = +4, or -4. (4) 
Subtracting (2) from (1), vey =8. (5) 
Dividing (5) by (4), x—y = +2, or -2. (6) 
From (4) and (6), oe 3, or —3; y =1, or -1. 
By substitution in (1) and (2) we find that the two pairs { ae 
and i % is satisfy (1) and (2). 
Example 2. Solve the system 
22 +7 = 16, (1) 
y? = 62. (2) 
SotuTion: Substituting 6z for y? in (1), we obtain 
x? +62 = 16, 
or 4+ 62 — 16 =0. (3) 


Solving (3) by formula (Art. 164), 
jae 6 + V96 + 64 Bae = 2, or cao 


If x =2, we have y = +V/12 = +273. 
If x = —8, wefindy =+ 4/—48, = + 41v/3, which are imaginary numbers. 
r=2 z=2 fra 3 
The possible solutions are then = 2 
on Pe Set have 


ey a ., each of which should be tested by actual substitution. 
y= 4iv/3 


EXERCISES 
Solve and check: 
1 OX, eae = 20; 
v—y+8 =0. rin IA Wh 
2. 2? + y? = 85, 4, 5x2? — 9y? + 121 = 0, 


oot a) ocecnad A 7y? — 32? — 105 = 0. 


262 EQUATIONS INVOLVING QUADRATICS [Cuar. XXIV. - 


Dee ya), 9. x? + 4peaee, 
ry = —7, v—ayty =3. 
6.27 Pay =36, 10. (x + y)? + (x — y)? — 50 = 0, 
xy +y*? = 45. (x —y)(a@+y) +7 =0. 
Ten det ry een Li Li) 2 yee 
sy+y’ =6. Ty sate 
8 p—g=9, 12. 2° — aio 
4p? = 25¢. : DY reas 


Hint: Divide the members of one 
equation by the corresponding mem- 
bers of the other. 


13. 2? — 7? = 1, 16. x? + dry + 2y? = 15, 

x—y =3. r+y =3. 

14. x7 — y? = 25, 17. 2m? + mn — n? = 2, 

x+y = 10. 2m —n=1. 

15. 2? — xy — 6y? = 16, 18. a? +7 = 4ab + 5b?, 
x + 2y = 16. QO? a= jeans 


MISCELLANEOUS PROBLEMS 


1. The sum of two numbers is 5 and the sum of their squares 
is 143. Find the numbers. 

2. A rectangular field 68,200 square feet in area, is sur- 
rounded by a 4-wire fence ; 5840 feet of wire were required 
for the fence. Find the dimensions of the field. 

3. The sum of the squares of two numbers is $2, and their 
product minus @ is equal to their difference. What are the 
numbers? 

4. The hypotenuse of a right triangle is 25 feet. The sum 
of the other two sides is 35 feet. What are the lengths of the 
sides? 

5. The area of a rectangle is 50 and the perimeter is 54. 
What are the dimensions of the rectangle? 





i Art. 174] PROBLEMS 263 


6. A piece of wire 48 inches long is bent into the form of 
a right triangle in which the hypotenuse is 20 inches long. Find 
the other sides of the triangle. 

7. A group of students club together to rent a suite of 
rooms for $400 per year. By adding 5 new members to the 
group the assessment was $4 less per member. How many 
students were there in the club at first? 

g. A certain number of two figures, when multiplied by 
the left digit, becomes 54; but when multiplied by the right 
digit, it becomes 189. What is the number? 

9. The annual income from an investment is $72. If the 
principal were $240 more and the rate of interest 1% less the 
income would remain unchanged. What are the principal and 
the rate of interest? 

10. A sum of money placed at simple interest for 6 years 
amounts to $5580. Had the interest been increased 1% it 
would have amounted to $45 more than this in 5 years. What 
are the principal and the rate of interest? 

11. A rectangular field is 145 yards long and 50 yards 
wide. How much must the width be increased and the length 
decreased in order that the area remain unchanged while the 
perimeter is decreased 30 yards? 

12. A certain kind of cloth loses 5% in width and 4% in 
length by shrinking. Find the length and width of a rec- 
tangular piece of the cloth whose shrinkage in area is 2.2 square 
yards and in perimeter 2.1 yards? 

13. An 8 by 10 photograph is enlarged until it covers twice 
the original area, keeping the ratio of the length to the width 
unchanged. Find the sides of the enlarged photograph. 

14. A man loaned $9000 in two unequal sums at such rates 
that both sums yielded the same annual interest. The larger 
sum at the higher rate of interest would yield $250 per year, 
and the smaller sum at the lower rate, $160 per year. How 
was the money divided and what were the rates of interest? 


264 EQUATIONS INVOLVING QUADRATICS [Cuap. XXIV. - 


15. A dealer sells a number of books for $1125, receiving — 
the same price for each book. If he had sold 150 books less 
but charged 25 cents more, he would have received the same 
sum. Find the price and the number of books. 


16. A dealer purchased a certain number of sheep for 
$175 ; after losing two of them he sold the rest at $2.50 a head 
more than he gave for them, and by so doing gained $5 by the 
deal. Find the number of sheep purchased. 


17. Both the numerator and the denominator of a frac- 
tion are increased by their squares. The new fraction reduces 
to 3. If both the numerator and the denominator are divided 
by their squares the result is 3. What is the fraction? 


18. The differences between the hypotenuse of a right tri- 
angle and the other two sides are } and 1 respectively. Find 
the sides of the triangle. 


19. The area and the perimeter of a rectangle are both 
25. What are the dimensions of the rectangle? 


20. A farmer is cutting wheat in a field which is twice as 
long as it is wide. After cutting a strip 10 rods wide around 
the outside of the field he estimates that 4 of the work has been 
done. What are the dimensions of the field? 





Art. 174] EXERCISES AND PROBLEMS 265 


REVIEW EXERCISES AND PROBLEMS 


Extract the following square roots : 


: Ves, Dap, ~/9-16, V9 +16, Vat + 20% +B. 
= 20n? +25 | /16- 100 - 36, VJ (a? +y)? —4(2? +y) +4. 
or +3)? me 
: : — 1 a oY 
3. Simplify: W/4a, ~/8a3, 3 Vi Ve V 45a?b’. 


4. If sis the side of a triangle having equal sides, what is the altitude? 
What is the area? ; 





SOLUTION: 
The altitude, h, bisects the base AC. 
B 
st. Ss* 
OR ee 5 a “3: — SS 
Then. At = 8 (5) s aieg 
| 382s 
2 i s 
ane h 4 5V3. 
We have then for the area, 
213 3? J 
Area = 5-5V3 = 7 V3. A Ds C 


5. The side of a triangle having equal sides, is 10 inches. Find the 
altitude and the area, correct to two decimal places. 

6. Find to two decimal places the side of an equilateral triangle - 
whose area is 100 square inches. 

7. Solve the equation A =r? forr. Find r if A = 50 square inches. 


8. Add 





ee and y -, and express the result as a fraction 
24375 4-V5 
with a rational denominator. 

9. Make up quadratic equations with the unknown x so that the 
values of x shall be (a) positive integers; (b) positive fractions; (c) negative 
integers ; (d) negative fractions ; (¢) both zero ; (f) one zero, and one a 
positive integer. 

10. Solve for x the equation 22? + 5a -4 =0. Calculate the roots to 
two decimal places. 

11. If the diameter of a circle is increased by 3 feet the area is doubled. 
Find the diameter, correct to two decimal places. 


266 EQUATIONS INVOLVING QUADRATICS [Cuap. XXIV. 


12. The perimeter of a rectangle is 82 inches. The diagonal is 29 inches. 
How long are the sides? 
13. The hypotenuse of a right triangle is 20 inches. If the altitude is 


multiplied by ~/2 and the base by +/3, the hypotenuse is multiplied by $. | 
Find the base and the altitude. 


a+b : b4 
a + ab op) (a eae 
14. Simplify > . 
a 
: (So tg | 


15. What number added to the denominators of ; and “ respectively 


will make the results equal? Under what condition is a solution impossi- 
ble? 

16. Plot the loci of the following equations and determine the solution 
of each pair from the graphs : 


y = 6, ee, ps ee 
Oe eee (0) 2x +y = 10; ©) — dy =3. 








17. Solve the equation s = ut + $f? for t. Find ¢ when wu = 10,f = 32.2, 
and s = 200. 


18. Plot the locus of the equation y = x? for values of x from 1 to 10. 
Then solve the equation for x. This gives x = ~/y. If now we choose a 
value of y to be 4, and read off from the graph the corresponding value 
of x, which is 2, we have the square root of the chosen value of y. Thus 
determine from the graph, as accurately as possible, the square root of each 
of the following: 16, 25, 36, 7, 2, 3, 60, 42. 


19. Which is the larger 1/3 or SE 


| V/11 aan 
20. Find the values of ——~——— to three significant figures (1) b 
ite i gures (1) by 
using the square roots of 5, 7, and 11; (2) by first rationalizing the denom- 
inator. 





21. Evaluate to three significant figures — 
V7 -2 





Abscissa, 180 

Absolute value, 25 

Addition, 
associative law for, 56 
commutative law for, 56 
of fractions, 143 
of monomials, 54 

_ of polynomials, 57 

of radicals, 229 





of signed numbers, 28 
on the number scale, 23 
Ahmes, 80, 137 
Algebraic expressions, 14 
Antecedent, 169 
Arabic notation, 13 
Axes of coérdinates, 180 


’ Base of a power, 9 
Binomial, 53 
cube of, 105 
square of, 100 
Braces, 15 
Brackets, 15 


Cancellation, 141 
Checking, 

an operation, 57 

a solution, 45 
Circle, 

area, 7 

circumference, 7 
Coefficient, 9 
Completing the square, 239, 240 
Complex fractions, 151 


INDEX 


[The numbers refer to pages.] 


Consequent, 169 
Constant, 175 
Coérdinates, 181 
Cube, 
of a binomial, 105 
of a number, 10 


Denominator, 39, 136 

rationalization of, 233 
Descartes, 27, 80, 181 
Difference, 

of two cubes, 119 

of two squares, 113 
Distributive law for, 73 
Dividend, 38 8 
Division, 38, 86 

by zero, 138 

law of exponents-for, 86 

of fractions, 149 

of monomials, 86 

of polynomials, 87, 89 

of quadratic surds, 2382 

of signed numbers, 38 

rule of exponents for, 86 

rule of signs for, 38 
Divisor, 38 

greatest common, 132 


Elimination, 194 
by addition and subtraction, 
195 
by substitution, 197 
Equality, 42 
members of an, 42 


268 


Equation, 43 
clearing of fractions, 159 
graph of an, 186 
historical note on, 80 
involving fractions, 81, 159 
involving parentheses, 78 
involving radicals, 235 
linear in two unknowns, 187 
linear or simple, 96 
literal, 164, 205 
locus of, 186, 187 
principles used in solving, 44 
quadratic, 126 
solution or root of, 44 
solved by factoring, 126, 127, 
238 
Equations, 
dependent or equivalent, 193 
graphical solution of, 187 
inconsistent, 193 
independent, 193 
simultaneous, 193 
solution of simultaneous linear, 
188, 193 
system of linear, 194 
system of quadratic, 256 
Evaluation 
of expressions, 16 
Exponent, 9 
Exponents, 
law for division, 86 
law for multiplication, 69 
Expression, 14 
terms of an, 53 
Extremes, 
of a proportion, 170 


\ 


Factor, 9, 109, 110 
common, 132 
found by grouping, 112 
highest common, 132 
integral, 110 


ae 


INDEX 


Factor, monomial, 110, 111 
of trinomials, 116, 118 
prime, 109, 110 
rational and integral, 110 
rationalizing, 234 
Factoring, 109 
equations solved by, 126, 127, 
238 
summary of, 121 
Fractions, 39, 136 
addition and subtraction of, 
143 
clearing equations of, 159 
complex, 151 
division of, 149 
equations involving, 81 
historical note on, 137 
lowest terms of, 140 
multiplication of, 145 
reduction to common denom- 
inator, 142 
reduction to lowest terms, 140 
signs in, 138 
square root of, 227 
terms of, 39, 136 
Function, 183 
graph of, 184, 247 


Graph, 

of a function, 184, 247 

of an equation, 186 
Graphical meaning, 

of imaginary roots, 250 — 
Graphical representation, 19, 180 

historical note on, 181 

of positive and negative num- 

bers, 23, 24 

of scientific data, 189 
Graphical solution, 

of equations, 187 
Greater than, 25 


INDEX 


_ Highest common factor, 132 
’ Historical note, 


on fractions, 137 

on graphical representation, 181 
on negative numbers, 27 

on symbols, 12 

on the equation, 80 


Identity, 42 


— Imaginary numbers, 249 


_ Index of a root, 226 


_ Integral expression, 110 


Irrational numbers, 226 


_ Less than, 25 


Linear equations, 
in one unknown, 96 
in three or more unknowns, 206 
in two unknowns, 187 
locus of, 187 
standard form in two unknowns, 
. 199 
systems of, 194, 206 
Locus, 
of a linear equation, 187 
of an equation, 186 
Lowest common multiple, 134 


Mean proportional, 171 
Means, 

of a proportion, 170 
Members of an equality, 42 


~ Monomials, 53 


addition of, 54 

division of, 86 

factors of, 110 

product of, 70 

square roots of, 214 

subtraction of, 60 
Multiple, 

common, 134 

lowest common, 134 


Multiplication, 69 


269 


associative and commutative 


laws of, 70 
by a monomial, 72 
distributive law of, 73 
in arithmetic, 35 
law of exponents for, 69 
of a product, 72 
of fractions, 145 
of monomials, 70 
of polynomials, 72, 75 
of powers, 69 
of quadratic surds, 231 
of signed numbers, 35 
rule of signs for, 36 
signs of, 2 


Negative numbers, 24 
historical note on, 27 
Numbers, 


graphical representations of, 


23, 24 
imaginary, 249 
irrational, 226 
positive and negative, 24 
rational, 226 
signed, 25 

Numerals, 

Arabic, 13 
Roman, 13 
Numerator, 39, 136 
Numerical value, 25 


Order of operations, 14 
Ordinate, 180 
Origin of codrdinates, 180 


Parentheses, 
equations involving, 78 
‘forms of, 15 
insertion of, 66 
removal of, 64 
uses of, 15 


270 INDEX 
Polynomials, 53 Proportional, 
addition of, 57 fourth, 172 
arrangement according to as- mean, 171 
cending and _ descending third, 171 
powers, 57 


division of, 87, 89 
factors of, 111 
multiplication of, 72, 75 
simplifying, 56 
subtraction of, 61 
Positive numbers, 24 
Power, 9 
Powers, 
ascending and descending, 57 
product of, 69 
Prime, 
expressions prime to each other, 
132 
factor, 109, 110 
Product, 35 
cross, 118 
of a polynomial by a mono- 
mial, 72 
of monomials, 70 
of polynomials, 75 
of powers, 69 
of the sum and difference, 102 
of two binomials with a com- 
mon term, 104 
Products, 
important type, 100 
Proportion, 170 
by addition, 173 
by addition and _ subtraction, 
174 
by alternation, 173 
by composition, 173 
by composition and division 
174 
by division, 174 
by inversion, 173 
by subtraction, 174 


Quadratic equations, 126, 127, 238 
in two unknowns, 256 ; 
solved by completing the 

square, 239 
solved by factoring, 127, 238 
solved by formula, 242 
special forms of, 245 
type form of, 242 

Quadratic surd, 227 
division of, 232 
multiplication of, 231 

Quadratic trinomial, 118 

Quotient, 38 
of monomials, 86 
of polynomials, 87, 89 


Radicals, 226 
addition and subtraction of, 
229 
division of, 232 
equations involving, 235 
multiplication of, 231 
similar, 229 
simplification of, 227 
Radical sign, 214 
Radicand, 228 
Ratio, 169 
Rational, 
expression, 109, 110 
number, 226 
Rationalization, 
of the denominator, 233 
Rationalizing factor, 234 
Reciprocal, 153 
Reduction, 
of fractions to lowest terms, 140 | 
to common denominator, 142 


INDEX 


* seght triangle, 80 


~ Roman numerals, 13 


_ Scale, 


in representing numbers, 23 
Signed numbers, 25 

addition of, 28 

division of, 38 

multiplication of, 35 

subtraction of, 29 
| Signs, 

in fractions, 138 

rule of for division, 38 


: rule of for multiplication, 36 
| imilar triangles, 178 
| fmaneon equations, 193 


of second degree, 256 
. olution, 
‘| ofan equation, 44 
of a pair of linear equations, 
188, 192 
of simultaneous quadratics, 257 
(quare, 
-\ completing the, 239 
of a binomial, 100 
of a number, 10 
of a trinomial, 106 
trinomial, 114 
uare root, 214 
of decimals, 221 
of fractions, 227 
of monomials, 214 
of numbers expressed in Arabic 
figures, 219 
of polynomials, 217 
of trinomials, 216 
process of finding a, pa 
ubscripts, 165 
tbtraction, 29 
of fractions, 143 


271 


of monomials, 60 

of polynomials, 61 

of radicals, 229 

of signed numbers, 29 

on the number scale, 23, 30 
rules for, 31 


Sum, 


of signed numbers, 28 
of two cubes, 119 


Surd, 226 


division of, 232 
multiplication of, 231 
quadratic, 227 


Symbols of operation, 1 


historical note on, 12 


System, 


of equations involving quad- 
ratics, 256 
of linens equations, 194 


Term, 53 


Similar or like, 53 


Transposition, 46 
Trinomial, 53 


general quadratic, 118 
Square, 114 

Square of a, 106 
Square root of a, 216 


Unknown, 44 


Variable, 175, 183 
Variation, 175 
Vineulum, 15 


Zero, 


division by, 138 
origin of, 13 








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